mm-2809 ==== Subject: Re: Update: Objections to Cantor's Theory > [standard argument for covering a countable subset > of R with artibtrarily small measure] So the real line has length less than e, where >I can choose e to be as small as I want. Oops. >> Doesn't e/2^n as n->infinity go to zero? How does zero >> cover anything? Oops. Well, yeah. That's why the sum of e/2^n from n=1..infinity > makes sense. Are you deliberately misreading, or just > stupid? >> Are you stupid or do you realize that infinite sums can be >> made to equal any value and they can even be made to >> diverge. > You are claiming that because some of them >> don't, there is more reals than naturals. That would also >> imply that because some of them do, there aren't more >> reals than naturals. No, that wouldn't follow. If A implies B, it doesn't follow that not A implies not B ; it > follows > that not B implies not A. >> That's not what's going on. I'm claiming that by your same >> reasoning, when all the reals are covered by a countable number >> of segments, we conclude that there aren't more reals. Get it >> now? I'm sorry, but that's exactly how you're (mis)using the previous > argument. A: The sequence of sums converges to a finite value. > B: There are more reals than naturals. A -> B is what was proven. You are saying wait, this can't be right, because consider... not A: The sequence of sums does not converge. > not B: There are not more reals than naturals. then by the same reasoning, we must have not A -> not B; so not B should follow from not A. >> No. Not converging does not mean diverging. > I didn't say it did (although a sum of strictly positive reals either > diverges or converges); I said that there is no reason to think that, > if I believe the OP's arguemnt is valid, that then by the same > reasoning, if all the reals are covered by a countable number of > segments, there are not more reals than naturals. >> But I >> do mean that the sum of a countable number of segments >> that diverge should make *YOU* (not me) think that there >> are fewer reals, because *YOU* (not I) think that somehow >> matters. > Suppose we have proven that If x is an integer, then x is rational. > You are saying that knowing that x is /not/ an integer should make *ME* > (not you) think that x is /not/ rational, because *I* (not you) think > that matters. > Why on earth would I think that? If x is not an integer, it could > either be rational or not. And in the proof in question, if the > sequence is one that covers all the reals and doesn't converge, then we > could still have the reals either more or the same number of > elements. > What matters is that the OP chose a sequence that converged, and used > that to prove something. > There is no reason for me to think that if the OP had chosen a > different sequence that did /not/ converge, that the proof would > therefore result in the /opposite/ conclusion. > That's the fallacy of A -> B, therefore not A -> not B. If you think its reasonable that a countable number of covers can't cover all the reals is insufficient to prove that there are more reals, then you are certainly a candidate for believing that if a countable number of covers can cover the reals, the reals are countable. In other words, if you accept one piece of nonsense, then its logical that you are willing to accept another piece of nonsense that follows a similar line of incorrect reasoning. That is logical. ==== Subject: Re: Update: Objections to Cantor's Theory >> That's not what's going on. I'm claiming that by your same >> reasoning, when all the reals are covered by a countable number >> of segments, we conclude that there aren't more reals. Get it >> now? I'm sorry, but that's exactly how you're (mis)using the previous > argument. A: The sequence of sums converges to a finite value. > B: There are more reals than naturals. A -> B is what was proven. You are saying wait, this can't be right, because consider... not A: The sequence of sums does not converge. > not B: There are not more reals than naturals. then by the same reasoning, we must have not A -> not B; so not B should follow from not A. >> No, But I >> do mean that the sum of a countable number of segments >> that diverge should make *YOU* (not me) think that there >> are fewer reals, because *YOU* (not I) think that somehow >> matters. > Suppose we have proven that If x is an integer, then x is rational. > You are saying that knowing that x is /not/ an integer should make *ME* > (not you) think that x is /not/ rational, because *I* (not you) think > that matters. > Why on earth would I think that? If x is not an integer, it could > either be rational or not. And in the proof in question, if the > sequence is one that covers all the reals and doesn't converge, then we > could still have the reals either more or the same number of > elements. B, therefore not A -> not B. > If you think its reasonable that a countable number of covers > can't cover all the reals is insufficient to prove that there are > more reals, Where did I assert that convoluted sentence? Since a countable number of covers CAN cover the reals, I suppose it's a reasonable as saying If 6 were 9, then it would be insuffcient to prove anything; logically true, but meaningless. > that if a countable number of covers can cover the reals, the > reals are countable. Hmm, kind of nice tongue twister. Anyway, it's obvious that a countable number of covers CAN cover all the reals; in fact the entire number line is a single cover that covers all the reals. There's nothing about the cardinality of the reals that can be deduced from this. Go back and read the original argument again. It says that IF we assume the reals are countable, THEN there would have to be a particular countable set of intervals (i) that would cover all the reals, and (ii) whose total length is finite. Since the real number line is infinite in length, (i) and (ii) together are a contradiction; so the assumption that the reals are countable must be false. This has nothing to do with any argument which states that IF we have a countable cover that (i) covers (or doesn't cover) the reals and (ii) whose total length is not finite, THEN the reals are not (or are) countable. > In other words, if you accept one piece of nonsense, then its > logical that you are willing to accept another piece of nonsense > that follows a similar line of incorrect reasoning. Well sure; but you still haven't said what part of the original argument you think is nonsense. > That is logical. It's logical, once you have shown that the original reasoning was incorrect. ==== Subject: Re: Update: Objections to Cantor's Theory >> That's not what's going on. I'm claiming that by your same >> reasoning, when all the reals are covered by a countable number >> of segments, we conclude that there aren't more reals. Get it >> now? I'm sorry, but that's exactly how you're (mis)using the previous > argument. A: The sequence of sums converges to a finite value. > B: There are more reals than naturals. A -> B is what was proven. You are saying wait, this can't be right, because consider... not A: The sequence of sums does not converge. > not B: There are not more reals than naturals. then by the same reasoning, we must have not A -> not B; so not B should follow from not A. > No, But I >> do mean that the sum of a countable number of segments >> that diverge should make *YOU* (not me) think that there >> are fewer reals, because *YOU* (not I) think that somehow >> matters. Suppose we have proven that If x is an integer, then x is rational. > You are saying that knowing that x is /not/ an integer should make *ME* > (not you) think that x is /not/ rational, because *I* (not you) think > that matters. Why on earth would I think that? If x is not an integer, it could > either be rational or not. And in the proof in question, if the > sequence is one that covers all the reals and doesn't converge, then we > could still have the reals either more or the same number of > elements. That's the fallacy of A -> B, therefore not A -> not B. >> If you think its reasonable that a countable number of covers >> can't cover all the reals is insufficient to prove that there are >> more reals, > Where did I assert that convoluted sentence? Since a countable number > of covers CAN cover the reals, I suppose it's a reasonable as saying > If 6 were 9, then it would be insuffcient to prove anything; > logically true, but meaningless. >> that if a countable number of covers can cover the reals, the >> reals are countable. > Hmm, kind of nice tongue twister. Anyway, it's obvious that a countable > number of covers CAN cover all the reals; in fact the entire number > line is a single cover that covers all the reals. There's nothing about > the cardinality of the reals that can be deduced from this. > Go back and read the original argument again. > It says that IF we assume the reals are countable, THEN there would > have to be a particular countable set of intervals (i) that would cover > all the reals, and (ii) whose total length is finite. Since the real > number line is infinite in length, (i) and (ii) together are a > contradiction; so the assumption that the reals are countable must be > false. > This has nothing to do with any argument which states that IF we have a > countable cover that (i) covers (or doesn't cover) the reals and (ii) > whose total length is not finite, THEN the reals are not (or are) > countable. >> In other words, if you accept one piece of nonsense, then its >> logical that you are willing to accept another piece of nonsense >> that follows a similar line of incorrect reasoning. > Well sure; but you still haven't said what part of the original > argument you think is nonsense. >> That is logical. > It's logical, once you have shown that the original reasoning was > incorrect. You said there was a problem with the logic. Now you're admitting its logical. That's all. You piped up, not about the math stuff, but about how A->B does not mean ~A->~B, remember? So what is it? Were you wrong about that and now you need to direct attention elsewhere before admitting my logic was ok? ==== Subject: Re: Update: Objections to Cantor's Theory >> In other words, if you accept one piece of nonsense, then its >> logical that you are willing to accept another piece of nonsense >> that follows a similar line of incorrect reasoning. > Well sure; but you still haven't said what part of the original > argument you think is nonsense. >> That is logical. > It's logical, once you have shown that the original reasoning was > incorrect. > You said there was a problem with the logic. There is a logical problem in your argument regarding Robert Low's proof, not in every single thing you say. > Now you're admitting > its logical. I agreed that it's logical (or at least sensible) that if you accept one piece of nonsense, that you might be willing to accept another. However, in order for this to be applied to Robert Low's argument, you need to show that the original argument is a piece of nonsense. You haven't done that. Is that clear enough? > That's all. You piped up, not about the math stuff, but > about how A->B does not mean ~A->~B, remember? Isn't understanding what constitutes a valid refutation of a mathematical argument math stuff? > So what is > it? Were you wrong about that and now you need to direct attention > elsewhere before admitting my logic was ok? Man, you've either totally lost track of your own argument, or you're approaching the Troll Zone. Way back when, you originally said: > Are you stupid or do you realize that infinite sums can be > made to equal any value and they can even be made to > diverge. You are claiming that because some of them > don't, there is more reals than naturals. That would also > imply that because some of them do, there aren't more > reals than naturals. the fallacy A->B implies ~A->~B. And it's still a fallacy. So your argument is still not valid; and Robert's argument is still not a piece of nonsense. ==== Subject: Re: Update: Objections to Cantor's Theory >> In other words, if you accept one piece of nonsense, then its >> logical that you are willing to accept another piece of nonsense >> that follows a similar line of incorrect reasoning. Well sure; but you still haven't said what part of the original > argument you think is nonsense. > That is logical. It's logical, once you have shown that the original reasoning was > incorrect. >> You said there was a problem with the logic. > There is a logical problem in your argument regarding Robert Low's > proof, not in every single thing you say. >> Now you're admitting >> its logical. > I agreed that it's logical (or at least sensible) that if you accept > one piece of nonsense, that you might be willing to accept another. > However, in order for this to be applied to Robert Low's argument, you > need to show that the original argument is a piece of nonsense. You > haven't done that. Is that clear enough? >> That's all. You piped up, not about the math stuff, but >> about how A->B does not mean ~A->~B, remember? > Isn't understanding what constitutes a valid refutation of a > mathematical argument math stuff? >> So what is >> it? Were you wrong about that and now you need to direct attention >> elsewhere before admitting my logic was ok? > Man, you've either totally lost track of your own argument, or you're > approaching the Troll Zone. > Way back when, you originally said: >> Are you stupid or do you realize that infinite sums can be >> made to equal any value and they can even be made to >> diverge. You are claiming that because some of them >> don't, there is more reals than naturals. That would also >> imply that because some of them do, there aren't more >> reals than naturals. > the fallacy A->B implies ~A->~B. And it's still a fallacy. So your > argument is still not valid; and Robert's argument is still not a piece > of nonsense. Since you isolated it to my paragraph (a good thing, since that's what I said) , you must either accept the premise that an infinite sum can be made to converge proves there are more reals than naturals, or you think that its nonsense to believe that the premise that an infinite sum converges proves there are more reals than naturals. No other outside information should matter. Don't try to distract. It's either something you accept or you don't. I don't. Therfore, Its logical to assume, in my opinion (as this is a matter of opinion), that if you don't believe the premise that an infintie sum converges proves there are more reals than naturals is nonsense, then you are probably (and I'm saying its reasonable to assume) willing to believe that the fact that an infinite sum diverges proves there are not more reals than naturals. To me, it seems like similar nonsense. Why are we going over this so many times? If you add additional arguments to try to convince me about the relative number of real numbers, that's a completely different argument and the discussion about the logic of a previous discussion has no bearing. So, suppose I agreed that you could establish enough facts to prove there are more reals than naturals. Then the fact that some nonsense was proposed and I concluded that people who bought that nonsense might buy similar nonsense is still logical. ==== Subject: Re: Update: Objections to Cantor's Theory > Man, you've either totally lost track of your own argument, or you're > approaching the Troll Zone. > Way back when, you originally said: >> Are you stupid or do you realize that infinite sums can be >> made to equal any value and they can even be made to >> diverge. You are claiming that because some of them >> don't, there is more reals than naturals. That would also >> imply that because some of them do, there aren't more >> reals than naturals. > the fallacy A->B implies ~A->~B. And it's still a fallacy. So your > argument is still not valid; and Robert's argument is still not a piece > of nonsense. Nothing that you say below actually addresses what I said above. I am not now, nor have I ever been, arguing with your statements to the effect that people who believe one kind of nonsense will often believe another; in fact I highly endorse this view. Usenet itself is the perfect repository of evidence in its favor. At any rate, your comments provoked these thoughts: (I found your sentence structure quite long for me to read, so I added brackets and some line breaks so as to clarify my understanding of your intent; no brackets occur in your original post. I apologize in advance if this offends.) > Since you isolated it to my paragraph (a good thing, since that's > what I said) , you must either accept the premise that > [an infinite sum can be made to converge proves there > are more reals than naturals], > or you think that its nonsense to believe the premise that > [an infinite sum converges proves there are more reals than > naturals]. Here's the thing. I don't simply accept the premise that ... because it seems reasonable to me, or because it fits my intuitions well, or that I believe in it in some religious way. Robert gave a PROOF of the fact that, from the accepted axioms, the premise is true. That's why I accept it - because it actually follows logically from the axioms. We can disagree about accepting a premise; but not about the conclusions that follow logically from the premises that we agree upon. > No other outside information should matter. Besides the proof, the outside information that matters here are the axioms and logistical system that are pre-agreed upon. Given these, his conclusion follows correctly; and so I automatically accept it because that is the mathematical sense of saying (something) is true. > Don't try > to distract. It's either something you accept or you don't. On the contrary, if you're doing mathematics, it's /not/ something you either accept or you don't. If you accept the axioms and don't accept the conclusion, then you're mathematically just plain wrong. If you don't accept the conclusion, then you can choose to reject the axioms. Your call! But mathematicians will still say that, if you accept the axioms, then such-and-such a conclusion is true. And if you just say hell, reject the axioms then!, there are other implications that come up. Can you still integrate a function? Is there still a number x such that x*x = 2? > I don't. Therefore, you aren't doing mathematics, you're just expressing an aesthetic opinion, as is your right. You're not saying I don't like the result; but I do like the axioms. Therefore Robert's proof must not be a proof. It must have a false step, and that false step is (some step in Robert's proof). And you're not saying I don't like this result; so let's reject the axiom of infinity. Then, I can prove that the result is no longer true; but it can still be proven that 2+2=4, etc.... Those would be mathematical responses to Robert. Instead you're saying I reject this logical argument because I simply don't accept its result can be true. Anything mathematical sounding you say from this kind of stance will be no more than that - mathematical sounding, but not actually making any kind of mathematical sense. And so it shouldn't surprise you that in response to your complaints, most mathematicians will then shrug their shoulders and respond so what?, or even hurl derisive abuse. > Therfore, > Its logical to assume, in my opinion (as this is a matter of opinion), > that if you don't believe the premise that > [an infinite sum converges proves there are more reals than naturals] > is nonsense, then you > are probably (and I'm saying its reasonable to assume) willing to > believe that > [the fact that an infinite sum diverges proves there > are not more reals than naturals]. > To me, it seems like similar nonsense. Your statement is one of opinion; as mathematics, it's literally non-sense. I read it as: Suppose we disagree on the truth of the statement P = Jessica Simpson is beautiful. Since we disagree, it's likely that we'll disagree on other things, especially similar things like Britny Spears is beautiful. That's fine as a statement about people and how they interact, but we were talking about Robert's proof - is it logically correct given the axioms and logistic structure we all agree on? It's not a matter of opinion. That's the kind of thing mathematicans talk about, and what they mean when they say things like there are more reals than naturals. > Why are we going over this so many times? Because you're not used to thinking in the way /mathematicians/ think about math. It sounds to me like you're more used to talking about philosophy than math. > If you add additional arguments to try to convince me about the > relative number of real numbers, that's a completely different > argument and the discussion about the logic of a previous > discussion has no bearing. Why try to convince you of something you've already made up your mind about? If you reject a correct logical argument on the basis that you don't like its conclusion, why should another correct logical argument yield any other response? The conclusion will still be the same. > So, suppose I agreed that you > could establish enough facts to prove there are more reals than > naturals. Then the fact that some nonsense was proposed and > I concluded that people who bought that nonsense might buy > similar nonsense is still logical. An insightful observation about people in general, but completely irrelevant as a mathematical argument. ==== Subject: Re: Update: Objections to Cantor's Theory [well, frankly, who cares what?] > Man, you've either totally lost track of your own argument, or you're > approaching the Troll Zone. A few days ago I came to the conclusion that he was firmly positioned under the bridge, just waiting for us poor gullible goats to trot over. Somehow his postings don't have the same ring of demented sincerity as those of certain other posters. ==== Subject: Re: Update: Objections to Cantor's Theory > that if a countable number of covers can cover the reals, the > reals are countable. > Hmm, kind of nice tongue twister. Anyway, it's obvious that a > countable number of covers CAN cover all the reals; in fact > the entire number line is a single cover that covers all the > reals. There's nothing about the cardinality of the reals that > can be deduced from this. > Go back and read the original argument again. > It says that IF we assume the reals are countable, THEN there > would have to be a particular countable set of intervals (i) > that would cover all the reals, and (ii) whose total length is > finite. Since the real number line is infinite in length, (i) > and (ii) together are a contradiction; so the assumption that > the reals are countable must be false. On the one hand, Robert Low's original argument looks bullet-proof, a countable set of points can be covered by a countable cover of arbitrarily small size. On the other hand, what happens if we apply the argument to the rationals? What follows are a series of reasonable-to-me assertions. I do believe there is a hole in there somewhere. I'm hoping for a little help in remembering something from some Once-Upon-A-Time math course. Assume we've got a list of the rationals, and we cover them with a sequence of open covers B_n of size e/2^n. Since the rationals Q are dense in R, the largest gap in the cover has to be of length zero containing a single irrational point. It seems to me that the maximum number of gaps in the cover would occur in the case of the intervals all being distinct, and, in that case, there would be a countable number of single-point gaps, which could be disposed of with another countable cover of arbitrarily small total length. This seems to say that the reals can be covered by a countable cover of arbitrarily small total length. Where did I go wrong? It seems to me that my trouble may come from mixing intuitions from integer-type order (as in the list of rationals) with intuitions from rational-type orders (as in placement on the real number line). Jim Burns ==== Subject: Re: Update: Objections to Cantor's Theory > On the one hand, Robert Low's original argument looks bullet-proof, > a countable set of points can be covered by a countable cover of > arbitrarily small size. That's on account of it being right. > On the other hand, what happens if we apply the argument to the > rationals? You have a proof that the rationals are a set of measure zero. It's a standard result in measure theory, and tells you that the rationals are simultaneously dense and somehow mysteriously sparse in the reals, both at the same time. Cool, eh? > Assume we've got a list of the rationals, and we cover them > with a sequence of open covers B_n of size e/2^n. Since the > rationals Q are dense in R, the largest gap in the cover > has to be of length zero containing a single irrational point. In general, intervals in the cover don't have a nearest neighbour, any more than rationals do. The intervals are really horribly mixed up. > It seems to me that my trouble may come from mixing intuitions > from integer-type order (as in the list of rationals) with > intuitions from rational-type orders (as in placement on the > real number line). Yep. ==== Subject: Re: Update: Objections to Cantor's Theory > On the one hand, Robert Low's original argument looks bullet-proof, > a countable set of points can be covered by a countable cover of > arbitrarily small size. > That's on account of it being right. It would be nice if being right were always enough. [...] > It seems to me that my trouble may come from mixing intuitions > from integer-type order (as in the list of rationals) with > intuitions from rational-type orders (as in placement on the > real number line). > Yep. By my count, I've gone back and forth three times in the hours since I last posted, flipping from being sure that I can describe how the rest of the reals show up to being sure that I can show that the rest of the reals are countable. I think I'm going to have to take this off-line and crack open an actual book. If you have any titles that you think treat this particular topic well, I would be grateful for them. Jim Burns ==== Subject: Re: Update: Objections to Cantor's Theory >>That's on account of it being right. > It would be nice if being right were always enough. Sometimes you have to settle for what you can get. > I think I'm going to have to take this off-line and crack > open an actual book. If you have any titles that you think > treat this particular topic well, I would be grateful > for them. I'm not sure. If you have access to a decent library, you might try scratching around in the measure theory section. I agree, it's hard to see how so many irrationals get mixed out (though it's 'obvious' that they are). I've tended to find that what I get out of this is an acceptance that the rationals and irrationals are mixed up in a way that's damned hard to get any good intuition for :-( ==== Subject: Re: Update: Objections to Cantor's Theory >> On the one hand, Robert Low's original argument looks bullet-proof, >> a countable set of points can be covered by a countable cover of >> arbitrarily small size. > That's on account of it being right. >> On the other hand, what happens if we apply the argument to the >> rationals? > You have a proof that the rationals are a set of measure > zero. It's a standard result in measure theory, and > tells you that the rationals are simultaneously dense > and somehow mysteriously sparse in the reals, both > at the same time. Cool, eh? Coolness is the property of being able to be dense and sparse. Sort of like the ostrich scientists' minds. Coolness is not an inconsistency, however. Coolness is a mathematical term that is well accepted by ostrich scientists to divert your attention away from an inconsistency. ==== Subject: Re: Update: Objections to Cantor's Theory >> On the one hand, Robert Low's original argument looks bullet-proof, >> a countable set of points can be covered by a countable cover of >> arbitrarily small size. > That's on account of it being right. On the other hand, what happens if we apply the argument to the >> rationals? > You have a proof that the rationals are a set of measure > zero. It's a standard result in measure theory, and > tells you that the rationals are simultaneously dense > and somehow mysteriously sparse in the reals, both > at the same time. Cool, eh? > Coolness is the property of being able to be dense and > sparse. Sort of like the ostrich scientists' minds. > Coolness is not an inconsistency, however. Coolness > is a mathematical term that is well accepted by ostrich > scientists to divert your attention away from an inconsistency. You were not asked if it is COOL to be COMMANDER-IN-CHIEF and issue orders. You were asked for your POSITION on the matter. Is that the considered judgement of this NG's ostrich-in-chief? karl m ==== Subject: Re: Update: Objections to Cantor's Theory > that if a countable number of covers can cover the reals, the > reals are countable. > Hmm, kind of nice tongue twister. Anyway, it's obvious that a >> countable number of covers CAN cover all the reals; in fact >> the entire number line is a single cover that covers all the >> reals. There's nothing about the cardinality of the reals that >> can be deduced from this. >> Go back and read the original argument again. >> It says that IF we assume the reals are countable, THEN there >> would have to be a particular countable set of intervals (i) >> that would cover all the reals, and (ii) whose total length is >> finite. Since the real number line is infinite in length, (i) >> and (ii) together are a contradiction; so the assumption that >> the reals are countable must be false. > On the one hand, Robert Low's original argument looks bullet-proof, > a countable set of points can be covered by a countable cover of > arbitrarily small size. > On the other hand, what happens if we apply the argument to the > rationals? > What follows are a series of reasonable-to-me assertions. > I do believe there is a hole in there somewhere. I'm hoping for a > little help in remembering something from some Once-Upon-A-Time > math course. > Assume we've got a list of the rationals, and we cover them > with a sequence of open covers B_n of size e/2^n. Since the > rationals Q are dense in R, the largest gap in the cover > has to be of length zero containing a single irrational point. > It seems to me that the maximum number of gaps in the cover > would occur in the case of the intervals all being distinct, > and, in that case, there would be a countable number of > single-point gaps, which could be disposed of with another > countable cover of arbitrarily small total length. > This seems to say that the reals can be covered by a countable > cover of arbitrarily small total length. Where did I go wrong? > It seems to me that my trouble may come from mixing intuitions > from integer-type order (as in the list of rationals) with > intuitions from rational-type orders (as in placement on the > real number line). > Jim Burns I'm certain that a ostrich scientist will steer you away from any line of thinking that would cause an inconsistency to be shown. ==== Subject: Re: Update: Objections to Cantor's Theory > I'm certain that a ostrich scientist will steer you away from any > line of thinking that would cause an inconsistency to be shown. Mr. Joker. Please address your remarks to the chair. The question on the floor is: Is that the considered judgement of this NG's ostrich-in-chief? karl m ==== Subject: Re: Update: Objections to Cantor's Theory [...] > This seems to say that the reals can be covered by a countable > cover of arbitrarily small total length. Where did I go wrong? > It seems to me that my trouble may come from mixing intuitions > from integer-type order (as in the list of rationals) with > intuitions from rational-type orders (as in placement on the > real number line). > I'm certain that a ostrich scientist will steer you away from any > line of thinking that would cause an inconsistency to be shown. Please show me that you are not an ostrich: You said in ; Which is it Cantorians, 1+omega=omega or 1+omega transcends ; omega? I said in : The original proof is enough that the sets of integers and : the reals are not the same size, which is all it needs to : show. : : However, there is an alternate version of the diagonal proof : that displays as many unlisted reals as there are both listed : and unlisted -- for any list. It uses the fact that reals have : both base ten and base seven expansions. Any base ten real, : if it avoids '0', '9', and the diagonal digit at every decimal : place, is guaranteed to be off the list. This leaves at least : seven choices at every decimal place, enough to map to /every/ : possibility at the corresponding heptimal place. : : So rather than the anti-Cantorians only needing to fit one : more real into Hilbert's Infinite Hotel, almost done, : they have as many reals left over as when they started, : not even begun. You say... ==== Subject: Re: Update: Objections to Cantor's Theory > [...] > This seems to say that the reals can be covered by a countable > cover of arbitrarily small total length. Where did I go wrong? It seems to me that my trouble may come from mixing intuitions > from integer-type order (as in the list of rationals) with > intuitions from rational-type orders (as in placement on the > real number line). >> I'm certain that a ostrich scientist will steer you away from any >> line of thinking that would cause an inconsistency to be shown. > Please show me that you are not an ostrich: > You said in > ; Which is it Cantorians, 1+omega=omega or 1+omega transcends > ; omega? > I said in > : The original proof is enough that the sets of integers and > : the reals are not the same size, which is all it needs to > : show. > : However, there is an alternate version of the diagonal proof > : that displays as many unlisted reals as there are both listed > : and unlisted -- for any list. It uses the fact that reals have > : both base ten and base seven expansions. Any base ten real, > : if it avoids '0', '9', and the diagonal digit at every decimal > : place, is guaranteed to be off the list. This leaves at least > : seven choices at every decimal place, enough to map to /every/ > : possibility at the corresponding heptimal place. > : So rather than the anti-Cantorians only needing to fit one > : more real into Hilbert's Infinite Hotel, almost done, > : they have as many reals left over as when they started, > : not even begun. > You say... I haven't put forth a theory and claimed it is completely consistent. I haven't denied apparent inconsistencies in mathematical theories. When you bring up an alternate version of the diagonal proof, can I assume you are agreeing that the original version is flawed? If not, why have the alternate? Let's clean up 1 mess at a time. If the original version is ok, then stop diverting attention away from it. If it's not ok, then admit to that. Let's suppose you have 2 flawed versions of a so-called proof that both have the same flawed conclusion. To keep you from using one of them to support the other, it seems reasonable to discuss one version only. Otherwise, you look like an ostrich. That's what I say. ==== Subject: Re: Update: Objections to Cantor's Theory > [...] > This seems to say that the reals can be covered by a countable > cover of arbitrarily small total length. Where did I go wrong? It seems to me that my trouble may come from mixing intuitions > from integer-type order (as in the list of rationals) with > intuitions from rational-type orders (as in placement on the > real number line). >> I'm certain that a ostrich scientist will steer you away from any >> line of thinking that would cause an inconsistency to be shown. > Please show me that you are not an ostrich: > You said in > ; Which is it Cantorians, 1+omega=omega or 1+omega transcends > ; omega? > I said in > : The original proof is enough that the sets of integers and > : the reals are not the same size, which is all it needs to > : show. > : > : However, there is an alternate version of the diagonal proof > : that displays as many unlisted reals as there are both listed > : and unlisted -- for any list. It uses the fact that reals have > : both base ten and base seven expansions. Any base ten real, > : if it avoids '0', '9', and the diagonal digit at every decimal > : place, is guaranteed to be off the list. This leaves at least > : seven choices at every decimal place, enough to map to /every/ > : possibility at the corresponding heptimal place. > : > : So rather than the anti-Cantorians only needing to fit one > : more real into Hilbert's Infinite Hotel, almost done, > : they have as many reals left over as when they started, > : not even begun. > You say... > I haven't put forth a theory and claimed it is completely consistent. > I haven't denied apparent inconsistencies in mathematical theories. What you have done is claim that Mainstream Math (or the Cantorians, if you like) is inconsistent because a particular proof didn't say what you wanted it to say: not just that Card(R) > Card(N) but by how much. I had run into an argument similar to yours before. (You may already be familiar with Herc, a schizophrenic from Townsville, Australia, who, in his spare time, is also God, Adam (upgraded version), and the subject of an worldwide conspiracy to beam voices into his head morning, noon, and night -- an altogether more reasonable person to deal with than you are.) Foolishly thinking that your argument was based on an honest misunderstanding (much as Herc's was), I sketched out a proof that showed you what you claimed you wanted: not only that Card(R) > Card, but by how much. (If f:N->R is any list of reals and R-f(N) is the set of the unlisted, then card(R-f(N))= Card(R). You're welcome.) > When you bring up an alternate version of the diagonal proof, > can I assume you are agreeing that the original version is flawed? You can assume that your understanding of the original version is flawed. > If not, why have the alternate? Let's clean up 1 mess at a time. And if the first mess is your doing? How long should I wait for you to clean it up? > If the original version is ok, then stop diverting attention away > from it. If it's not ok, then admit to that. I admit that you do not want to talk about a proof giving you what you claimed to want, only a few days ago. > Let's suppose you have 2 flawed versions of a so-called proof > that both have the same flawed conclusion. To keep you from > using one of them to support the other, it seems reasonable to > discuss one version only. Otherwise, you look like an ostrich. > That's what I say. Let's suppose you are a troll. ==== Subject: Re: Update: Objections to Cantor's Theory > When you bring up an alternate version of the diagonal proof, > can I assume you are agreeing that the original version is flawed? > If not, why have the alternate? Let's clean up 1 mess at a time. If > the original version is ok, then stop diverting attention away from > it. If it's not ok, then admit to that. There are, IIRC, over 100 distinct proofs of the Pythagorean theorem, even one by a President of the USA (unrelated, AFAIK, to the present one). None of these proofs, AFAIK, casts any doubt on the validity of any of the others. > Let's suppose you have 2 flawed versions of a so-called proof > that both have the same flawed conclusion. To keep you from > using one of them to support the other, it seems reasonable to > discuss one version only. Otherwise, you look like an ostrich. > That's what I say. And by saying so, PJ displays the IQ of an ostrich. Looks are not everything! ==== Subject: Re: Update: Objections to Cantor's Theory >> When you bring up an alternate version of the diagonal proof, >> can I assume you are agreeing that the original version is flawed? >> If not, why have the alternate? Let's clean up 1 mess at a time. If >> the original version is ok, then stop diverting attention away from >> it. If it's not ok, then admit to that. > There are, IIRC, over 100 distinct proofs of the Pythagorean theorem, > even one by a President of the USA (unrelated, AFAIK, to the present > one). None of these proofs, AFAIK, casts any doubt on the validity of > any of the others. >> Let's suppose you have 2 flawed versions of a so-called proof >> that both have the same flawed conclusion. To keep you from >> using one of them to support the other, it seems reasonable to >> discuss one version only. Otherwise, you look like an ostrich. >> That's what I say. > And by saying so, PJ displays the IQ of an ostrich. Looks are not > everything! I'm certain you make others in your profession proud to be like you. ==== Subject: Re: Update: Objections to Cantor's Theory >Between every pair of irrational numbers there is a rational umber. >This means that in fact there are not more irrationals than rationals. Would you mind filling in the argument, for those of us >>who have inadequate brains to do it for ourselves? >If these brains do not immediately grasp the meaning, then, I am >afraid, there is no hope that elaborate explanations will be of much >help. Nevertheless I will try my best: >If there were more irrationals than rationals in a linear order, then >at least two irrationals must exists without a rational between them. >>This is a complete nonsequitur; it's a 'paradox' that most >>of us get over when we're about 18. >>But lets assume that the irrationals are countable. Then >>the reals are countable. So let f:N -> R be a bijection from >>N to R. >>Now, let e be some small number (say, 10^-6). >>Let U_1 be the interval of width e, centred on f(1). >>Let U_2 be that of width e/2, centred on f(2). >>... >>U_n be that of width e/2^n, centred on f(n) >>... >>Then the union of all the U_n has total width >>at most 2e (because some of the intervals might >>overlap). >>But now I've covered all the reals, using a total >>length of 2e, where e can be as small as I like. >>So the real line has length less than e, where >>I can choose e to be as small as I want. Oops. > Doesn't e/2^n as n->infinity go to zero? How does zero > cover anything? Oops. Sure, e/2^n -> 0, but Sum(n=0..infinity) e/2^n = 2e ==== Subject: Re: Update: Objections to Cantor's Theory > If there were more irrationals than rationals in a linear order, then > at least two irrationals must exists without a rational between them. Non sequitor. Both the rationals and the irrations are dense in the set of real numbers. You are confounding cardinality and order again. Now put on your thinking cap. Reals = Rationals union Irrationals (two disjoint sets there) if the rationals are countable and the irrationals are countable then the reals being the union of two disjoint countable sets would be countable. But we know this is not the case from the diagnoal arguament. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory >> If there were more irrationals than rationals in a linear order, then >> at least two irrationals must exists without a rational between them. > Non sequitor. Both the rationals and the irrations are dense in the set of > real numbers. You are confounding cardinality and order again. > Now put on your thinking cap. Reals = Rationals union Irrationals (two > disjoint sets there) if the rationals are countable and the irrationals > are countable then the reals being the union of two disjoint countable > sets would be countable. But we know this is not the case from the > diagnoal arguament. So basically, the argument is that there must be more irrationals than rationals because otherwise there wouldn't be, right? ==== Subject: Re: Update: Objections to Cantor's Theory > So basically, the argument is that there must be more irrationals > than rationals because otherwise there wouldn't be, right? No. The argument is this. The assumption that the reals are coutable leads to a contradiction. This is the thrust of the diagonal argument. You are truly an intellectual vandal. If you cannot comprehend or master an idea, you cover it with your not so funny grafitti. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory >> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? > No. The argument is this. The assumption that the reals are coutable leads > to a contradiction. This is the thrust of the diagonal argument. For those who accept what you say as true, an acceptable conclusion may be that the axioms are faulty, right? ==== Subject: Re: Update: Objections to Cantor's Theory >> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? > No. The argument is this. The assumption that the reals are coutable leads > to a contradiction. This is the thrust of the diagonal argument. > For those who accept what you say as true, an acceptable conclusion > may be that the axioms are faulty, right? Not unless one can also show that the reals are countable. It takes a contradiction within the axiom system to show that system to be faulty, but nothing within the system finds uncountability to contradict anything WITHIN THE SYSTEM. That it may contradict things outside the system is irrelevant to the consistency of the system. ==== Subject: Re: Update: Objections to Cantor's Theory >> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? No. The argument is this. The assumption that the reals are coutable > leads > to a contradiction. This is the thrust of the diagonal argument. >> For those who accept what you say as true, an acceptable conclusion >> may be that the axioms are faulty, right? > Not unless one can also show that the reals are countable. It takes a > contradiction within the axiom system to show that system to be faulty, > but nothing within the system finds uncountability to contradict > anything WITHIN THE SYSTEM. > That it may contradict things outside the system is irrelevant to the > consistency of the system. The faulty logic here has already been pointed out elsewhere. ==== Subject: Re: Update: Objections to Cantor's Theory >> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? No. The argument is this. The assumption that the reals are coutable > leads > to a contradiction. This is the thrust of the diagonal argument. For those who accept what you say as true, an acceptable conclusion >> may be that the axioms are faulty, right? > Not unless one can also show that the reals are countable. It takes a > contradiction within the axiom system to show that system to be faulty, > but nothing within the system finds uncountability to contradict > anything WITHIN THE SYSTEM. > That it may contradict things outside the system is irrelevant to the > consistency of the system. > The faulty logic here has already been pointed out elsewhere. Conclusions which fault a logical system where no logical fault is known are not acceptable. ==== Subject: Re: Update: Objections to Cantor's Theory So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? No. The argument is this. The assumption that the reals are coutable > leads > to a contradiction. This is the thrust of the diagonal argument. >> For those who accept what you say as true, an acceptable conclusion >> may be that the axioms are faulty, right? Not unless one can also show that the reals are countable. It takes a > contradiction within the axiom system to show that system to be faulty, > but nothing within the system finds uncountability to contradict > anything WITHIN THE SYSTEM. That it may contradict things outside the system is irrelevant to the > consistency of the system. >> The faulty logic here has already been pointed out elsewhere. > Conclusions which fault a logical system where no logical fault is known > are not acceptable. George Bush is not acceptable. ==== Subject: Re: Update: Objections to Cantor's Theory > Conclusions which fault a logical system where no logical fault is known > are not acceptable. > George Bush is not acceptable. Then again, George Bush is not logical. ==== Subject: Re: Update: Objections to Cantor's Theory >> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? No. The argument is this. The assumption that the reals are coutable > leads > to a contradiction. This is the thrust of the diagonal argument. For those who accept what you say as true, an acceptable conclusion >> may be that the axioms are faulty, right? Not unless one can also show that the reals are countable. It takes a > contradiction within the axiom system to show that system to be faulty, > but nothing within the system finds uncountability to contradict > anything WITHIN THE SYSTEM. That it may contradict things outside the system is irrelevant to the > consistency of the system. The faulty logic here has already been pointed out elsewhere. > Conclusions which fault a logical system where no logical fault is known > are not acceptable. > George Bush is not acceptable. With that I concur. ==== Subject: Re: Update: Objections to Cantor's Theory >> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? No. The argument is this. The assumption that the reals are coutable > leads > to a contradiction. This is the thrust of the diagonal argument. For those who accept what you say as true, an acceptable conclusion >> may be that the axioms are faulty, right? Not unless one can also show that the reals are countable. It takes a > contradiction within the axiom system to show that system to be faulty, > but nothing within the system finds uncountability to contradict > anything WITHIN THE SYSTEM. That it may contradict things outside the system is irrelevant to the > consistency of the system. The faulty logic here has already been pointed out elsewhere. > Conclusions which fault a logical system where no logical fault is known > are not acceptable. > George Bush is not acceptable. Maybe that's why he's not here, with us? Can you relate this to your PROPOSITION? Do you need help from your SPONSOR? karl m ==== Subject: Re: Update: Objections to Cantor's Theory >> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? > No. The argument is this. The assumption that the reals are coutable > leads > to a contradiction. This is the thrust of the diagonal argument. For those who accept what you say as true, an acceptable conclusion >> may be that the axioms are faulty, right? > Not unless one can also show that the reals are countable. It takes a > contradiction within the axiom system to show that system to be faulty, > but nothing within the system finds uncountability to contradict > anything WITHIN THE SYSTEM. > That it may contradict things outside the system is irrelevant to the > consistency of the system. > The faulty logic here has already been pointed out elsewhere. I'm going to rule that this doesn't constitute MOTION on your part. karl m ==== Subject: Re: Update: Objections to Cantor's Theory >> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? > No. The argument is this. The assumption that the reals are coutable leads > to a contradiction. This is the thrust of the diagonal argument. > For those who accept what you say as true, an acceptable conclusion > may be that the axioms are faulty, right? No. Axioms cannot be faulty as a conclusion. As Virgil said, they only speak after being called-into-play during the proof process. We agree to axioms after deciding that the resulting SYSTEM is consistent, has been consistent, and therefore will always be consistent. karl m ==== Subject: Re: Update: Objections to Cantor's Theory >> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? No. The argument is this. The assumption that the reals are coutable > leads > to a contradiction. This is the thrust of the diagonal argument. >> For those who accept what you say as true, an acceptable conclusion >> may be that the axioms are faulty, right? > No. Axioms cannot be faulty as a conclusion. As Virgil said, they only > speak after being called-into-play during the proof process. > We agree to axioms after deciding that the resulting SYSTEM is > consistent, has been consistent, and therefore will always be > consistent. karl m Here's a system with no inconsistencies: AXIOM 1: All Cantorians live in fantasyland. Note that no proofs derived from this system results in an inconsistency. Furthermore, there is a good model right here on sci.logic. ;-) ==== Subject: Re: Update: Objections to Cantor's Theory >> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? No. The argument is this. The assumption that the reals are coutable > leads > to a contradiction. This is the thrust of the diagonal argument. For those who accept what you say as true, an acceptable conclusion >> may be that the axioms are faulty, right? > No. Axioms cannot be faulty as a conclusion. As Virgil said, they only > speak after being called-into-play during the proof process. > We agree to axioms after deciding that the resulting SYSTEM is > consistent, has been consistent, and therefore will always be > consistent. karl m > Here's a system with no inconsistencies: > AXIOM 1: All Cantorians live in fantasyland. > Note that no proofs derived from this system results in an inconsistency. > Furthermore, there is a good model right here on sci.logic. ;-) Since Fantasyland, i.e., the imagination, is where all mathematics takes place, anyway, it would be an ideal place for mathematicians to live. Unfortunately for them, their bodies, at least, are trapped in a far more mundane land. ==== Subject: Re: Update: Objections to Cantor's Theory > Here's a system with no inconsistencies: > AXIOM 1: All Cantorians live in fantasyland. > Note that no proofs derived from this system results in an inconsistency. > Furthermore, there is a good model right here on sci.logic. ;-) > Since Fantasyland, i.e., the imagination, is where all mathematics > takes place, anyway, it would be an ideal place for mathematicians to > live. Unfortunately for them, their bodies, at least, are trapped in a > far more mundane land. There's a great name for a theme park Far More Mundaneland!!!. Probably worth a visit by people with a lot of turmoil in their real life. ==== Subject: Re: Update: Objections to Cantor's Theory > Since Fantasyland, i.e., the imagination, is where all mathematics > takes place, anyway, it would be an ideal place for mathematicians to > live. Unfortunately for them, their bodies, at least, are trapped in a > far more mundane land. How true. ==== Subject: Re: Update: Objections to Cantor's Theory ual.net> <6ByGe.1418$Zh.187@tornado.rdc-kc.rr.com Since Fantasyland, i.e., the imagination, is where all mathematics > takes place, anyway, it would be an ideal place for mathematicians to > live. Unfortunately for them, their bodies, at least, are trapped in a > far more mundane land. > How true. This consists of YET_ANOTHER_OPINION. You were asked to PROVIDE MOTION. karl m ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> <3ks78uFvhhrvU1@individual.net> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? > No. The argument is this. The assumption that the reals are coutable leads > to a contradiction. This is the thrust of the diagonal argument. > For those who accept what you say as true, an acceptable conclusion > may be that the axioms are faulty, right? If one requires a theory in which there are no uncountable sets, then, yes, one would need a different set of axioms. But if one does not require that the theory preclude uncountable sets, then the axioms are not made faulty by their providing uncountable sets. Meanwhile, if one does require a theory in which there are no uncountable sets, or no infinite sets, or no sets larger than a certain natural number, then one would give axioms for such a theory. MoeBlee ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> <3ks78uFvhhrvU1@individual.net> uncountable sets, or no infinite sets, or no sets larger than a certain > natural number, then one would give axioms for such a theory. > MoeBlee Here's my question (probably obvious to people more familiar with this stuff than I am): Suppose we take ZF (no Choice!), and replace the Axiom of Infinity with the statement that there exists a set W which can be placed into bijection with a proper subset of itself. Does it follow that the set of naturals exists? I'm guessing not, because the elements of W are too indistinguishable to be ordered - without Choice, how do I choose some element, and then another different element, etc.? But then, wadda I know? ==== Subject: Re: Update: Objections to Cantor's Theory > Suppose we take ZF (no Choice!), and replace the Axiom of Infinity with > the statement that there exists a set W which can be placed into > bijection with a proper subset of itself. > Does it follow that the set of naturals exists? > I'm guessing not, because the elements of W are too indistinguishable > to be ordered - without Choice, how do I choose some element, and then > another different element, etc.? But then, wadda I know? intitial guess is that you're right. The axiom of infinity doesn't just give an infinite set; it gives an inductive set starting with an already defined constant (viz. 0), which is something more specific. Even with the axiom of choice, how can you derive the axiom of infinity from the proposed new axiom? Would you use oridinals and recursion on them? Would you be able to define a UNIQUE set? (I've only given this a brief moment, so maybe it would be easier than I imagine.) Is this in context of mollifying objectors to set theory? I don't think they like bijection with a proper subset any more than they like the axioms of infinity or choice. I'm interested in hearing more about what you have in mind. MoeBlee ==== Subject: Re: Update: Objections to Cantor's Theory ual.net> the statement that there exists a set W which can be placed into > bijection with a proper subset of itself. > Does it follow that the set of naturals exists? > I'm guessing not, because the elements of W are too indistinguishable > to be ordered - without Choice, how do I choose some element, and then > another different element, etc.? But then, wadda I know? I'm self-taught, so I know a little bit about a lot things, but not a more set theory than me. > Anyway, my > intitial guess is that you're right. The axiom of infinity doesn't just > give an infinite set; it gives an inductive set starting with an > already defined constant (viz. 0), which is something more specific. > Even with the axiom of choice, how can you derive the axiom of infinity > from the proposed new axiom? Would you use oridinals and recursion on > them? When I idly posted this, I thought that sprinkling the magical pixie dust of Choice would somehow make this too easy, but now I can't even see a way of getting there with Choice. The whole point of the AoI seems to be to ensure, as you say, that _some_ kind of recursion indeed will end up yielding an actual set, w. A similar thought was to note that W (my axiomatic infinite set) has a bijection to some proper subset W'. Then it seems easy to prove (waves hands) that then W' has a bijection to some proper subset W'', etc., literally ad infinitum. But that's no different than the Peano construction of the naturals; you still need something like the AoI to ensure that the set {W, W', W'', ...} (and thus N in its usual meaning) actually exists. > Would you be able to define a UNIQUE set? (I've only given this a > brief moment, so maybe it would be easier than I imagine.) What I was hoping for was something along the lines of because W is infinite, it must contain a subset which has the same order relation as the naturals, and then appeal to AoC to select the set from the set of those subsets of W satisfying that premise. > Is this in context of mollifying objectors to set theory? I don't think > they like bijection with a proper subset any more than they like the > axioms of infinity or choice. Oh yes, I live to mollify the objectors. Sleep, genetle objectors, sleep; yes, cease your jabbber and sleep... :-) No, but for me (being self taught), I really learn a lot when I try to figure out the clearest way to describe why the axioms of, say, ZFC actually are sensible, useful rules. It makes sure I stay honest! And there's always the possibility that the lightbulb will go on in someone's head. At the same time, of course ZFC is all that is good and true (flashes official Evil Cantorian membership card), but I try to take all complaints seriously. By bending the rules, you can see where the chosen rules come from. > I'm interested in hearing more about what you have in mind. Well, to amplify off the top of my head, suppose we can construct an ordering on W (specify a subset of W x W) such that: 1) trichotomy holds (for all x,y in W, exactly one of xy). 2) for all x, exists y with x < y. 3) let M(x,y) = {z in W : x What I was hoping for was something along the lines of because W is > infinite, it must contain a subset which has the same order relation as > the naturals, and then appeal to AoC to select the set from the set > of those subsets of W satisfying that premise. There's no need for choice. Suppose f is a bijection between M and its proper subset M', and let a be an element in MM'. f then has the properties of a successor function, with a as 0. The corresponding set {a,f(a),f(f(a)),...} is defined as the intersection of all subsets A of M that contain a and are closed under f. To get from this a set containing the empty set and closed under the function x -> x U {x} we need to use replacement. ==== Subject: Re: Update: Objections to Cantor's Theory ual.net> its proper subset M', and let a be an element in MM'. f then has > the properties of a successor function, with a as 0. The corresponding > set {a,f(a),f(f(a)),...} is defined as the intersection of all > subsets A of M that contain a and are closed under f. ==== Subject: Re: Update: Objections to Cantor's Theory > Meanwhile, if one does require a theory in which there are no > uncountable sets, or no infinite sets, or no sets larger than a certain > natural number, then one would give axioms for such a theory. > MoeBlee > Here's my question (probably obvious to people more familiar with this > stuff than I am): > Suppose we take ZF (no Choice!), and replace the Axiom of Infinity with > the statement that there exists a set W which can be placed into > bijection with a proper subset of itself. > Does it follow that the set of naturals exists? I'm sorry, but the chair has already ruled that Mr. Joker's motion is not framed in the form of a motion. karl m ==== Subject: Re: Update: Objections to Cantor's Theory > For those who accept what you say as true, an acceptable conclusion > may be that the axioms are faulty, right? Wrong. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory >> For those who accept what you say as true, an acceptable conclusion >> may be that the axioms are faulty, right? > Wrong. So I take it you think any set of given axioms are always the correct ones to be using? ==== Subject: Re: Update: Objections to Cantor's Theory > So I take it you think any set of given axioms are always the correct > ones to be using? Any consistent mathematical theory is alright or correct. For specific purposes some mathematical theories are more usefal than others. The only thing that can be wrong about a mathematical theory is that it be internally inconsistent. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory >> So I take it you think any set of given axioms are always the correct >> ones to be using? > Any consistent mathematical theory is alright or correct. For > specific purposes some mathematical theories are more usefal than others. > The only thing that can be wrong about a mathematical theory is that it be > internally inconsistent. You didn't answer. You used scare quotes to make it seem like you have a good answer. ==== Subject: Re: Update: Objections to Cantor's Theory So I take it you think any set of given axioms are always the correct >> ones to be using? > Any consistent mathematical theory is alright or correct. For > specific purposes some mathematical theories are more usefal than others. > The only thing that can be wrong about a mathematical theory is that it be > internally inconsistent. > You didn't answer. Since PJ is so good at reading between the lines anyway, he should be able to read that he has been anwered in the negative when it is as obvious as in this instance.. ==== Subject: Re: Update: Objections to Cantor's Theory >> So I take it you think any set of given axioms are always the correct >> ones to be using? Any consistent mathematical theory is alright or correct. For > specific purposes some mathematical theories are more usefal than > others. > The only thing that can be wrong about a mathematical theory is that it > be > internally inconsistent. >> You didn't answer. > Since PJ is so good at reading between the lines anyway, he should be > able to read that he has been anwered in the negative when it is as > obvious as in this instance.. In other words I was right when I said that it may be that the axioms are faulty. > ==== Subject: Re: Update: Objections to Cantor's Theory > So I take it you think any set of given axioms are always the >> correct >> ones to be using? Any consistent mathematical theory is alright or correct. For > specific purposes some mathematical theories are more usefal than > others. > The only thing that can be wrong about a mathematical theory is that > it > be > internally inconsistent. >> You didn't answer. Since PJ is so good at reading between the lines anyway, he should be > able to read that he has been anwered in the negative when it is as > obvious as in this instance.. >> In other words I was right when I said that it >> may be that the axioms are faulty. > Wrong, as usual. An axiom system may be inappropriate without being the > least bit faulty, so if one has the wrong axiom system for a particular > purpose, so no one need think along such peculiar lines that PJ tries to > impose on others. Ostrich scientist as usual. Wouldn't want to agree, right? That's part of being in the ostrich science too, isn't it? You obviously agree with the concept so you disagree with the exact wording because otherwise you would have to admit that you didn't realize that the original statement was correct. You must hide from ever being incorrect. This whole thread is about the ostrich science, the incoherence, and specifically how these things relate to Cantor's so-called diagonal proof. ==== Subject: Re: Update: Objections to Cantor's Theory >> . > So I take it you think any set of given axioms are always the >> correct >> ones to be using? Any consistent mathematical theory is alright or correct. For > specific purposes some mathematical theories are more usefal than > others. > The only thing that can be wrong about a mathematical theory is that > it > be > internally inconsistent. You didn't answer. Since PJ is so good at reading between the lines anyway, he should be > able to read that he has been anwered in the negative when it is as > obvious as in this instance.. In other words I was right when I said that it >> may be that the axioms are faulty. > Wrong, as usual. An axiom system may be inappropriate without being the > least bit faulty, so if one has the wrong axiom system for a particular > purpose, so no one need think along such peculiar lines that PJ tries to > impose on others. > Ostrich scientist as usual. Is that the considered judgement of this NG's ostrich-in-chief? ==== Subject: Re: Update: Objections to Cantor's Theory So I take it you think any set of given axioms are always the >> correct >> ones to be using? Any consistent mathematical theory is alright or correct. For > specific purposes some mathematical theories are more usefal than > others. > The only thing that can be wrong about a mathematical theory is that > it > be > internally inconsistent. You didn't answer. Since PJ is so good at reading between the lines anyway, he should be > able to read that he has been anwered in the negative when it is as > obvious as in this instance.. In other words I was right when I said that it >> may be that the axioms are faulty. > Wrong, as usual. An axiom system may be inappropriate without being the > least bit faulty, so if one has the wrong axiom system for a particular > purpose, so no one need think along such peculiar lines that PJ tries to > impose on others. > Ostrich scientist as usual. Wouldn't want to agree, right? That's > part of being in the ostrich science too, isn't it? You obviously > agree with the concept so you disagree with the exact wording > because otherwise you would have to admit that you didn't > realize that the original statement was correct. You must hide from > ever being incorrect. > This whole thread is about the ostrich science, the incoherence, and > specifically how these things relate to Cantor's so-called diagonal > proof. The AGENDA is set by the chair at the beginning of the session. THERE IS NO POSITION TO CRITICIZE THE CHAIR'S ORDERING OF THE AGENDA, outside the political commissar who confers with chair before the session. karl m ==== Subject: Re: Update: Objections to Cantor's Theory So I take it you think any set of given axioms are always the correct >> ones to be using? Any consistent mathematical theory is alright or correct. For > specific purposes some mathematical theories are more usefal than > others. > The only thing that can be wrong about a mathematical theory is that it > be > internally inconsistent. You didn't answer. > Since PJ is so good at reading between the lines anyway, he should be > able to read that he has been anwered in the negative when it is as > obvious as in this instance.. > In other words I was right when I said that it > may be that the axioms are faulty. Wrong, as usual. An axiom system may be inappropriate without being the least bit faulty, so if one has the wrong axiom system for a particular purpose, so no one need think along such peculiar lines that PJ tries to impose on others. ==== Subject: Re: Update: Objections to Cantor's Theory <3knfrmFv4gfkU2@individual.net> So I take it you think any set of given axioms are always the correct >> ones to be using? Any consistent mathematical theory is alright or correct. For > specific purposes some mathematical theories are more usefal than > others. > The only thing that can be wrong about a mathematical theory is that it > be > internally inconsistent. You didn't answer. > Since PJ is so good at reading between the lines anyway, he should be > able to read that he has been anwered in the negative when it is as > obvious as in this instance.. > In other words I was right when I said that it > may be that the axioms are faulty. If a PROBATIONARY MEMBER is unhappy with his ASSIGNED SPONSOR, one can be reassigned, resources permitting. Do you have a choice in this matter? karl m ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> 413$Zh.643@tornado.rdc-kc.rr.com> For those who accept what you say as true, an acceptable conclusion >> may be that the axioms are faulty, right? > Wrong. > So I take it you think any set of given axioms are always the correct > ones to be using? That depends on your CONSTITUTION. There is a process to change it, but it requires CONCENSUS from both the OPEN AND CLOSED sections of the organization simultaneously. karl m ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> <3ks78uFvhhrvU1@individual.net> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? > No. The argument is this. The assumption that the reals are coutable leads > to a contradiction. This is the thrust of the diagonal argument. > For those who accept what you say as true, an acceptable conclusion > may be that the axioms are faulty, right? No. If you start with a self-consistent system (of axioms) and adding an additional assumption leads to a contradiction, it is the additional assumption which is at fault. That is the basis of proof by contradiction. Your interpretation says that as soon as any proof by contradiction reaches the point of contradiction, you should suspect the axioms. - Randy ==== Subject: Re: Update: Objections to Cantor's Theory >> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? No. The argument is this. The assumption that the reals are coutable > leads > to a contradiction. This is the thrust of the diagonal argument. >> For those who accept what you say as true, an acceptable conclusion >> may be that the axioms are faulty, right? > No. > If you start with a self-consistent system (of axioms) > and adding an additional assumption leads to a contradiction, > it is the additional assumption which is at fault. > That is the basis of proof by contradiction. Your > interpretation says that as soon as any proof by > contradiction reaches the point of contradiction, > you should suspect the axioms. I said may be. Read before you judge. ==== Subject: Re: Update: Objections to Cantor's Theory >> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? No. The argument is this. The assumption that the reals are coutable > leads > to a contradiction. This is the thrust of the diagonal argument. For those who accept what you say as true, an acceptable conclusion >> may be that the axioms are faulty, right? > No. > If you start with a self-consistent system (of axioms) > and adding an additional assumption leads to a contradiction, > it is the additional assumption which is at fault. > That is the basis of proof by contradiction. Your > interpretation says that as soon as any proof by > contradiction reaches the point of contradiction, > you should suspect the axioms. > I said may be. Read before you judge. I did. Since that is not an acceptable conclusion, your statement that it may be is incorrect, just as in examining the ancient proof by contradiction that sqrt(2) is irrational, an acceptable conclusion may NOT be that the starting axioms are incorrect. - Randy ==== Subject: Re: Update: Objections to Cantor's Theory So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? No. The argument is this. The assumption that the reals are coutable > leads > to a contradiction. This is the thrust of the diagonal argument. >> For those who accept what you say as true, an acceptable conclusion >> may be that the axioms are faulty, right? No. If you start with a self-consistent system (of axioms) > and adding an additional assumption leads to a contradiction, > it is the additional assumption which is at fault. That is the basis of proof by contradiction. Your > interpretation says that as soon as any proof by > contradiction reaches the point of contradiction, > you should suspect the axioms. >> I said may be. Read before you judge. > I did. > Since that is not an acceptable conclusion, your > statement that it may be is incorrect, just as in > examining the ancient proof by contradiction that > sqrt(2) is irrational, an acceptable conclusion > may NOT be that the starting axioms are incorrect. It is a true statement that: You are a moron if you think that axioms have never changed due to undesirable consequences that were not inconsistencies. Therefore, some people may conclude that the axioms may be faulty. This is due to the fact that *SOME* people might feel that axioms that generate undesirable consequences are faulty. I didn't mention *WHO* might be making the conclusions and certainly I didn't count anyone out. So it is certainly plausable that what I said is true. ==== Subject: Re: Update: Objections to Cantor's Theory >> So basically, the argument is that there must be more irrationals >> than rationals because otherwise there wouldn't be, right? No. The argument is this. The assumption that the reals are coutable > leads > to a contradiction. This is the thrust of the diagonal argument. For those who accept what you say as true, an acceptable conclusion >> may be that the axioms are faulty, right? No. If you start with a self-consistent system (of axioms) > and adding an additional assumption leads to a contradiction, > it is the additional assumption which is at fault. That is the basis of proof by contradiction. Your > interpretation says that as soon as any proof by > contradiction reaches the point of contradiction, > you should suspect the axioms. I said may be. Read before you judge. > I did. > Since that is not an acceptable conclusion, your > statement that it may be is incorrect, just as in > examining the ancient proof by contradiction that > sqrt(2) is irrational, an acceptable conclusion > may NOT be that the starting axioms are incorrect. > It is a true statement that: > You are a moron if you think that axioms have never > changed due to undesirable consequences that were > not inconsistencies. Are you over 18? If not we'll need a ruling from a higher body granting you an extension of the appropriate sections to cover your presence. > Therefore, some people may conclude that the axioms > may be faulty. This is due to the fact that *SOME* > people might feel that axioms that generate undesirable > consequences are faulty. > I didn't mention *WHO* might be making the conclusions > and certainly I didn't count anyone out. So it is certainly > plausable that what I said is true. Again, the chair has called for you to ADVANCE YOUR MOTION TO THE FLOOR. karl m ==== Subject: Re: Update: Objections to Cantor's Theory > Would you mind filling in the argument, for those of us > who have inadequate brains to do it for ourselves? He is using WM logic. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory >What does 'more reals than naturals' mean? You, like many others, >seem to be assuming there is some 'true' definition of more. Most english speaking non-mathemeticians understand the meaning of the words they use, so yes, we admit to such knowledge. > In the context of set theory 'more' is not well defined, but > informally many people will say B has more elements than A > if there does not exist a surjective mapping from A to B. Don't forget that the side-effect of doing so is an incoherent stance that allows you to flip-flop. > If you mean something else by 'more', you have to define precisely > what you mean by it. Do mathemeticians need help with simple word meanings? Apparently so. > If you accept that 'more' means there does not exist > a surjection between two sets, then Cantor did indeed > prove that the are more reals than naturals. If you > think 'more' means something else, then describe > what you mean by 'more' mathematically. Describe what you mean by if mathematically. Describe what you mean by you mathematically. Describe what you mean by accept mathematically. Describe what you mean by that mathematically. Describe what you mean by the mathematically. Describe what you mean by are mathematically. etc. ==== Subject: Re: Update: Objections to Cantor's Theory >>What does 'more reals than naturals' mean? You, like many others, >>seem to be assuming there is some 'true' definition of more. > Most english speaking non-mathemeticians understand the meaning > of the words they use, so yes, we admit to such knowledge. Then pray tell what the 'true' definition of more is. What does it mean for a set to have more elements than another set? Stephen ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> What does it mean for a set to have more elements than >another set? Apparently it depends on one's profession but since mathematicians make up a minority of all people, its silly to think that any meaning they make up should be accepted by all people. Undoubtedly when non-mathematicians are called cranks because they disagree with arcane, and admittedly (by mathematicians) informal meanings of common words, then mathematicians are the people that are wrong. ==== Subject: Re: Update: Objections to Cantor's Theory georgie says... >Apparently it depends on one's profession but since mathematicians >make up a minority of all people, its silly to think that any meaning >they make up should be accepted by all people. Tell me who, besides a mathematician, care less about the definition of set size for infinite sets? Yes, maybe you do, but I'm willing to make a bet that the number of non-mathematicians who have an opinion about how to define size for infinite sets is much, much smaller than the number of mathematicians. I've never actually *met* such a person. (Although USENET provides evidence that they exist.) -- Daryl McCullough Ithaca, NY ==== Subject: Re: Update: Objections to Cantor's Theory > georgie says... >>Apparently it depends on one's profession but since mathematicians >>make up a minority of all people, its silly to think that any meaning >>they make up should be accepted by all people. > Tell me who, besides a mathematician, care less about the > definition of set size for infinite sets? > Yes, maybe you do, but I'm willing to make a bet that the number > of non-mathematicians who have an opinion about how to define > size for infinite sets is much, much smaller than the number of > mathematicians. I've never actually *met* such a person. (Although > USENET provides evidence that they exist.) > -- I realize the junk that mathaticians do is totally worthless outside their own little world, but non-mathematicians have their own logical theories that remain consistent with the rest of the world and that includes the word meanings that the entire rest of the world uses. If mathematicians need to call people cranks because of that, then the mathematicians are morons. ==== Subject: Re: Update: Objections to Cantor's Theory > I realize the junk that mathaticians do is totally worthless outside > their own little world, The junk that mathematicians do makes physics possible. Starting with caclculus which Newton invented to quantify motion. Without symmetry groups, Hilbert Space, Hermitean Operators there would be no quantum theory. Without Reimannian junk such as tensors there would be no General Theory of Relativity (and no GPS to boot). The theory of real and complex variables with its associated vector spaces, manifolds and such like is absolutely necessary to do physics. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory Poker Joker says... >I realize the junk that mathaticians do is totally worthless outside >their own little world, but non-mathematicians have their own >logical theories... Non mathematicians generally have no opinion about the relative sizes of infinite sets. It just doesn't come up, outside of mathematics. The number of people who have an opinion (and one that is different from that of mathematicians) is completely negligible. As I said, I've never personally met such a person. -- Daryl McCullough Ithaca, NY ==== Subject: Re: Update: Objections to Cantor's Theory > Poker Joker says... >>I realize the junk that mathaticians do is totally worthless outside >>their own little world, but non-mathematicians have their own >>logical theories... > Non mathematicians generally have no opinion about > the relative sizes of infinite sets. Shows what you know. BTW - Size is not a formally defined or widely accepted term. I think you are talking about cardinality. But of course I'm not a mathematician and the term never comes up in my discussions (according to you). > It just > doesn't come up, outside of mathematics. The number of > people who have an opinion (and one that is different from > that of mathematicians) is completely negligible. Is that why these threads get so long and have so many people involved? > As I > said, I've never personally met such a person. I can see why they keep their distance. You think they are negligible. > -- ==== Subject: Re: Update: Objections to Cantor's Theory Poker Joker says... >> Non mathematicians generally have no opinion about >> the relative sizes of infinite sets. >Shows what you know. I know plenty of non mathematicians. I'm married to one. Not a single one of them has ever expressed an opinion about the relative sizes of infinite sets. -- Daryl McCullough Ithaca, NY ==== Subject: Re: Update: Objections to Cantor's Theory On 27 Jul 2005 19:56:27 -0700, stevendaryl3016@yahoo.com (Daryl >Poker Joker says... > Non mathematicians generally have no opinion about > the relative sizes of infinite sets. >>Shows what you know. >I know plenty of non mathematicians. I'm married to one. >Not a single one of them has ever expressed an opinion >about the relative sizes of infinite sets. Most people, if asked, will offer an opinion about almost anything they understand or think they understand. As far as relative sizes of infinite sets, for those who are aware of the concept of infinity. but unaware of the mathematical concept of cardinality for infinite sets, the 2 most common opinions, based on my experience, are these: (1) If an infinite set A is a proper subset of infinite set B, then B has the larger size. So for example, the set of positive integers has a larger size than the set of even positive integers. (2) All infinite sets have equal size. What's infinity minus 1? Infinity, right? Hence there's no point comparing size for infinite sets. All infinite sets have the same size -- infinity. In fact, before Cantor, most mathematicians would probably would have agreed with one of the above 2 views, at least informally. But at the same time, I'm sure most would have pointed out that there was no formal definition of size for infinite sets, and many would have insisted that no such definition was possible. Cardinality is only one measure of size for infinite sets. Mathematicians understand that. Cantor understood that. For well ordered sets, ordinal numbers provide a finer measure than cardinal numbers. The definition of a size concept for sets has to satisfy a few simple requirements: (1) (counting principle) For finite sets, size is based on the number of elements. In other words, if you can count the number of elements, that's the size. (2) (size of a subset) If A is a subset of B, then A is less than or equal in size to B. (3) (transitivity) If A is less than or equal in size to B and B is less than or equal in size to C then A is less than or equal in size to C. (4) (weak anti-symmetry) If A is less than or equal in size to B and B is less than or equal in size to A, then A has the same size as B. Other than the above requirements, a size concept for sets is arbitrary, except, don't call it size, except informally, so as not to cause confusion. Motivate it as a kind of size and make sure to prove that it satisfies (1), (2), (3), (4), but call it something else. quasi ==== Subject: Re: Update: Objections to Cantor's Theory > Most people, if asked, will offer an opinion about almost anything > they understand or think they understand. Opinions are like ass-holes. Everybody has at least one. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory > Poker Joker says... > Non mathematicians generally have no opinion about > the relative sizes of infinite sets. >>Shows what you know. > I know plenty of non mathematicians. I'm married to one. > Not a single one of them has ever expressed an opinion > about the relative sizes of infinite sets. That's because they don't like being called negligible. ==== Subject: Re: Update: Objections to Cantor's Theory > Poker Joker says... > Non mathematicians generally have no opinion about > the relative sizes of infinite sets. Shows what you know. > I know plenty of non mathematicians. I'm married to one. > Not a single one of them has ever expressed an opinion > about the relative sizes of infinite sets. > That's because they don't like being called negligible. To call the number of a group negligible is not the same as calling any of the individuals in that group negligible. The number of Popes in the Roman Catholic Church is negligible in comparison to the number of Roman Catholics. That hardly makes the Pope negligible in any sense. ==== Subject: Re: Update: Objections to Cantor's Theory >I realize the junk that mathaticians do is totally worthless outside >their own little world, but non-mathematicians have their own >logical theories that remain consistent with the rest of the world >and that includes the word meanings that the entire rest of the world >uses. If mathematicians need to call people cranks because of >that, then the mathematicians are morons. It's exactly because the rest of the world does *not* agree on the word meanings it uses or on the logic that connects these meanings, that mathematicians (perhaps starting with Euclid) took up the task of trying to make things precise. By first clarifying the basic notions of logic and proof, then separating out undefined terms and axioms, proceeding to definitions, and then proving theorems with sufficient detail that any mathematician in the world could then verify the correctness (or incorrectness) of the proof, the arguments that plague the rest of the world are avoided (until the rest of the world, jealous of the logical paradise we have, barges in and tries to claim it's worthless). quasi ==== Subject: Re: Update: Objections to Cantor's Theory >>I realize the junk that mathaticians do is totally worthless outside >>their own little world, but non-mathematicians have their own >>logical theories that remain consistent with the rest of the world >>and that includes the word meanings that the entire rest of the world >>uses. If mathematicians need to call people cranks because of >>that, then the mathematicians are morons. > It's exactly because the rest of the world does *not* agree on the > word meanings it uses or on the logic that connects these meanings, > that mathematicians (perhaps starting with Euclid) took up the task of > trying to make things precise. By first clarifying the basic notions > of logic and proof, then separating out undefined terms and axioms, > proceeding to definitions, and then proving theorems with sufficient > detail that any mathematician in the world could then verify the > correctness (or incorrectness) of the proof, the arguments that plague > the rest of the world are avoided (until the rest of the world, > jealous of the logical paradise we have, barges in and tries to claim > it's worthless). > quasi Let's not hide behind formalism here. We are talking about a word that has been admitted by mathematicians in this thread to have no formal mathematical definition. ==== Subject: Re: Update: Objections to Cantor's Theory >I realize the junk that mathaticians do is totally worthless outside >their own little world, but non-mathematicians have their own >logical theories that remain consistent with the rest of the world >and that includes the word meanings that the entire rest of the world >uses. If mathematicians need to call people cranks because of >that, then the mathematicians are morons. >> It's exactly because the rest of the world does *not* agree on the >> word meanings it uses or on the logic that connects these meanings, >> that mathematicians (perhaps starting with Euclid) took up the task of >> trying to make things precise. By first clarifying the basic notions >> of logic and proof, then separating out undefined terms and axioms, >> proceeding to definitions, and then proving theorems with sufficient >> detail that any mathematician in the world could then verify the >> correctness (or incorrectness) of the proof, the arguments that plague >> the rest of the world are avoided (until the rest of the world, >> jealous of the logical paradise we have, barges in and tries to claim >> it's worthless). >> quasi >Let's not hide behind formalism here. We are talking about a word that >has been admitted by mathematicians in this thread to have no formal >mathematical definition. What word is that? ==== Subject: Re: Update: Objections to Cantor's Theory >I realize the junk that mathaticians do is totally worthless outside >their own little world, but non-mathematicians have their own >logical theories that remain consistent with the rest of the world >and that includes the word meanings that the entire rest of the world >uses. If mathematicians need to call people cranks because of >that, then the mathematicians are morons. It's exactly because the rest of the world does *not* agree on the >> word meanings it uses or on the logic that connects these meanings, >> that mathematicians (perhaps starting with Euclid) took up the task of >> trying to make things precise. By first clarifying the basic notions >> of logic and proof, then separating out undefined terms and axioms, >> proceeding to definitions, and then proving theorems with sufficient >> detail that any mathematician in the world could then verify the >> correctness (or incorrectness) of the proof, the arguments that plague >> the rest of the world are avoided (until the rest of the world, >> jealous of the logical paradise we have, barges in and tries to claim >> it's worthless). quasi >Let's not hide behind formalism here. We are talking about a word that >has been admitted by mathematicians in this thread to have no formal >mathematical definition. > What word is that? It doesn't matter what the SPECIFIC word is today. karl m ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> What does it mean for a set to have more elements than >another set? > Apparently it depends on one's profession Are there professions other than mathematics where the concept of comparing infinite set sizes come up? > but since mathematicians > make up a minority of all people, its silly to think that any meaning > they make up should be accepted by all people. The name made up by mathematicians should be accepted by people who want to discuss the mathematical concept. If that same concept arises elsewhere, say in international finance, the bankers are free to make up their own name for it. There is indeed no need for them to use the mathematician's name for the concept if they are going to converse only with other bankers, who use a different name. I agree that ordinary people have no need to accept mathematicians definitions for concepts which the ordinary people aren't discussing. So what? > Undoubtedly when > non-mathematicians are called cranks because they disagree with > arcane, and admittedly (by mathematicians) informal meanings of > common words, then mathematicians are the people that are wrong. Um, yes. It is crankish for a non-specialist to tell the professionals in any profession that some of their jargon is wrong, if it's the word they all use to describe an agreed-upon concept, one which the non-professional never has occasion to name. Do you have your own name for, say, non-blocking function call? Is that term as used in computer programming wrong because you may not understand what they mean by it? I submit that it is arcane and not understandable by non-programmers, but I also submit that if any person ever had occasion to write a non-blocking function call, they would also have no problem accepting that name for it. - Randy ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> What does it mean for a set to have more elements than >another set? > Apparently it depends on one's profession > Are there professions other than mathematics where the concept > of comparing infinite set sizes come up? Yes, there are some religious sects which, besides mathematicians, believe in the infinite. ==== Subject: Re: Update: Objections to Cantor's Theory Then pray tell what the 'true' definition of more is. >What does it mean for a set to have more elements than >another set? Apparently it depends on one's profession > Are there professions other than mathematics where the concept > of comparing infinite set sizes come up? > Yes, there are some religious sects which, besides mathematicians, > believe in the infinite. Non-responsive! Does WM suggest the existence of a PROFESSION which COMPARES infinite sizes, other than mathematics? ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> Then pray tell what the 'true' definition of more is. >What does it mean for a set to have more elements than >another set? Apparently it depends on one's profession > Are there professions other than mathematics where the concept > of comparing infinite set sizes come up? > Yes, there are some religious sects which, besides mathematicians, > believe in the infinite. Not an answer to my question. Can you provide a cite of a religious sect where the concept arises of comparing two infinite things and declaring one to be larger? But you bring up a second question: Do you not believe in infinite sets? - Randy ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> Then pray tell what the 'true' definition of more is. >What does it mean for a set to have more elements than >another set? Apparently it depends on one's profession Are there professions other than mathematics where the concept > of comparing infinite set sizes come up? > Yes, there are some religious sects which, besides mathematicians, > believe in the infinite. > Not an answer to my question. > Can you provide a cite of a religious sect where > the concept arises of comparing two infinite things and > declaring one to be larger? Maybe Christianity - Can God make a rock so big, that he himself cannot lift it? Or how about the Trinity - Is God bigger than the Holy Spirit? > But you bring up a second question: Do you not believe in > infinite sets? He seems to be of two minds on this (or perhaps more). > - Randy ==== Subject: Re: Update: Objections to Cantor's Theory > Yes, there are some religious sects which, besides mathematicians, > believe in the infinite. Is the set of integers finite or infinite? Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory >> Yes, there are some religious sects which, besides mathematicians, >> believe in the infinite. > Is the set of integers finite or infinite? Why do you care? A better question: Is the set of integers white or some other color? ==== Subject: Re: Update: Objections to Cantor's Theory > Why do you care? > A better question: > Is the set of integers white or some other color? Infinity applied as a predicate to sense is meaningful (A set is infinite if it can be put into 1-1 correspondence with a proper subset). Color on the other hand is not a predicate applicable to sets. Color is the frequency of light in the visible frequence range. It has nothing whatever to do with sets. So the question: what is the color of the set of integers is meaningless. While the question: is the set of integers infinite or not is meaningful. Sets do not posses color. They are not physical. They neither emit or reflect electromagnetic radiation. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory >> Why do you care? >> A better question: >> Is the set of integers white or some other color? > Infinity applied as a predicate to sense is meaningful (A set is infinite > if it can be put into 1-1 correspondence with a proper subset). Color on > the other hand is not a predicate applicable to sets. Color is the > frequency of light in the visible frequence range. It has nothing whatever > to do with sets. > So the question: what is the color of the set of integers is meaningless. > While the question: is the set of integers infinite or not is meaningful. > Sets do not posses color. They are not physical. They neither emit or > reflect electromagnetic radiation. I'm not aware of infinite integers as the Cantorians are, so no it doesn't make sense. ==== Subject: Re: Update: Objections to Cantor's Theory > I'm not aware of infinite integers as the Cantorians are, so no it doesn't > make sense. Each and every one of the integers is finite. It is the -set of integers- that is infinite. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory >> I'm not aware of infinite integers as the Cantorians are, so no it >> doesn't >> make sense. > Each and every one of the integers is finite. It is the -set of integers- > that is infinite. Why ask if you already have an answer that you like? ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> I'm not aware of infinite integers as the Cantorians are, so no it >> doesn't >> make sense. > Each and every one of the integers is finite. It is the -set of integers- > that is infinite. > Why ask if you already have an answer that you like? The process of REACHING A CONCENSUS requires that each participant say yes or no to the proposition as it stands; or they can reserve their opinion and submit them to one of the higher bodies, in which case they are still BOUND TO THE DECISION THEY DECLINED TO PARTICIPATE IN. karl m ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> Yes, there are some religious sects which, besides mathematicians, >> believe in the infinite. > Is the set of integers finite or infinite? > Why do you care? > A better question: > Is the set of integers white or some other color? It's a starting point for a discussion that leads to a diagnostic. When you find someone stuck in a circle you throw them a line. It's from naval intelligence, I guess. karl m ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> believe in the infinite. > Is the set of integers finite or infinite? No set is infinite, because the elements of a set must be distinguished by at least one property. For that sake one needs at least one part of reality. But the part(icle)s of reality are finite. All other belief is but that. (But do not accuse me of advocating in favour of a largest integer.) ==== Subject: Re: Update: Objections to Cantor's Theory Yes, there are some religious sects which, besides mathematicians, > believe in the infinite. > Is the set of integers finite or infinite? > No set is infinite, because the elements of a set must be distinguished > by at least one property. For that sake one needs at least one part of > reality. According to what definition in what axiom system do sets or their members have to conform to any physical standards? ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> believe in the infinite. Is the set of integers finite or infinite? > No set is infinite, because the elements of a set must be distinguished > by at least one property. For that sake one needs at least one part of > reality. > According to what definition in what axiom system do sets or their > members have to conform to any physical standards? You need at least one label to identify an element of a set, a number for instance. This label may be on a sheet of paper or in your brain. But it cannot exist without a part(icle) of physical reality repesenting it. So at least one proton in your brain must represent your idea. ==== Subject: Re: Update: Objections to Cantor's Theory > According to what definition in what axiom system do sets or their > members have to conform to any physical standards? > You need at least one label to identify an element of a set, a number > for instance. This label may be on a sheet of paper or in your brain. > But it cannot exist without a part(icle) of physical reality > repesenting it. So at least one proton in your brain must represent > your idea. Since the members of WM's sets do not have to be actual, but can be merely potential, their labels need not be actual either, until specifically called for. On the other hand, there is nothing in any axiom systems for any set theory other than WM's which requires labels. ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> Yes, there are some religious sects which, besides mathematicians, > believe in the infinite. > Is the set of integers finite or infinite? > No set is infinite, because the elements of a set must be distinguished > by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. > All other belief is but that. > (But do not accuse me of advocating in favour of a largest integer.) If you have concluded this, then you have concluded the set of integers is infinite. karl m ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> believe in the infinite. Is the set of integers finite or infinite? > No set is infinite, because the elements of a set must be distinguished > by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. > All other belief is but that. > (But do not accuse me of advocating in favour of a largest integer.) > If you have concluded this, then you have concluded the set of integers > is infinite. The values of integers are *potentially* infinite = not limited by any threshold. The set of integers is a bit more complicated than a simple mind may guess. There is no largest element. The set is not fixed. All we know is that it has less than 10^100 elements. See: Physical Constraints of Numbers, arxiv.org/abs/math.GM/0505649 ==== Subject: Re: Update: Objections to Cantor's Theory > The set of integers is a bit more complicated than a simple > mind may guess. There is no largest element. The set is not fixed. If it is a set in any standard set thory, then it is fixed. So whatever WM is preaching about, it is not sets. ==== Subject: Re: Update: Objections to Cantor's Theory >> The set of integers is a bit more complicated than a simple >> mind may guess. There is no largest element. The set is not fixed. > If it is a set in any standard set thory, then it is fixed. > So whatever WM is preaching about, it is not sets. We realized long ago that understanding simple things is beyond your capability. ==== Subject: Re: Update: Objections to Cantor's Theory >> The set of integers is a bit more complicated than a simple >> mind may guess. There is no largest element. The set is not fixed. > If it is a set in any standard set thory, then it is fixed. > So whatever WM is preaching about, it is not sets. > We realized long ago that understanding simple things > is beyond your capability. While it it possible that I will never understand such simple things as WM and PJ, I am can get along nicely without. ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> The set of integers is a bit more complicated than a simple >> mind may guess. There is no largest element. The set is not fixed. > If it is a set in any standard set thory, then it is fixed. > So whatever WM is preaching about, it is not sets. > We realized long ago that understanding simple things > is beyond your capability. WE meet each other face to face without fear of criticism. Kindly speak in the first person. karl m ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> mind may guess. There is no largest element. The set is not fixed. > If it is a set in any standard set thory, then it is fixed. > So whatever WM is preaching about, it is not sets. It is, unfortunately, not so naive a model as usual set theory. So I doubt you will understand it within the next 5 years. But it is correct and unavoidable. ==== Subject: Re: Update: Objections to Cantor's Theory > The set of integers is a bit more complicated than a simple > mind may guess. There is no largest element. The set is not fixed. > If it is a set in any standard set thory, then it is fixed. > So whatever WM is preaching about, it is not sets. > It is, unfortunately, not so naive a model as usual set theory. So I > doubt you will understand it within the next 5 years. But it is correct > and unavoidable. Whyever should any sane person ever want to understand it? And on what authority or proof does WM declare either the correctness or the unavoidability of his idiot ideas? WM has no mathematical justification for either correctness or unavoidability of his daydreams. ==== Subject: Re: Update: Objections to Cantor's Theory !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > The set of integers is a bit more complicated than a simple > mind may guess. There is no largest element. The set is not fixed. >> If it is a set in any standard set thory, then it is fixed. >> So whatever WM is preaching about, it is not sets. > It is, unfortunately, not so naive a model as usual set theory. You misspelled well-defined. > So I doubt you will understand it within the next 5 years. You have given no sign of understanding any of the gunk you purport to preach yourself: at least you keep contradicting yourself on it. -- ==== Subject: Re: Update: Objections to Cantor's Theory > The values of integers are *potentially* infinite = not limited by any > threshold. The set of integers is a bit more complicated than a simple > mind may guess. There is no largest element. The set is not fixed. All > we know is that it has less than 10^100 elements. How do we know that? ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> <3kutg1F1081p3U1@individual.net The values of integers are *potentially* infinite = not limited by any > threshold. The set of integers is a bit more complicated than a simple > mind may guess. There is no largest element. The set is not fixed. All > we know is that it has less than 10^100 elements. > How do we know that? ==== Subject: Re: Update: Objections to Cantor's Theory The values of integers are *potentially* infinite = not limited by any > threshold. The set of integers is a bit more complicated than a simple > mind may guess. There is no largest element. The set is not fixed. All > we know is that it has less than 10^100 elements. > How do we know that? Actually, it is the limited resources of WM's imagination that is the problem here. ==== Subject: Re: Update: Objections to Cantor's Theory comcast.com> <3kutg1F1081p3U1@individual.net> threshold. The set of integers is a bit more complicated than a simple > mind may guess. There is no largest element. The set is not fixed. All > we know is that it has less than 10^100 elements. How do we know that? > Actually, it is the limited resources of WM's imagination that is the > problem here. The standard is to remove the item from the agenda under the order you already gave. malbrain ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> <3kutg1F1081p3U1@individual.net The values of integers are *potentially* infinite = not limited by any > threshold. The set of integers is a bit more complicated than a simple > mind may guess. There is no largest element. The set is not fixed. All > we know is that it has less than 10^100 elements. > How do we know that? Apparently, from the report, no one has dared call for any more. It's only a matter of time until someone does. It's inevitable. karl m ==== Subject: Re: Update: Objections to Cantor's Theory !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> The values of integers are *potentially* infinite = not limited by >> any threshold. The set of integers is a bit more complicated than a >> simple mind may guess. There is no largest element. The set is not >> fixed. All we know is that it has less than 10^100 elements. > How do we know that? Hemp. -- ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> believe in the infinite. Is the set of integers finite or infinite? No set is infinite, because the elements of a set must be distinguished > by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. All other belief is but that. (But do not accuse me of advocating in favour of a largest integer.) > If you have concluded this, then you have concluded the set of integers > is infinite. > The values of integers are *potentially* infinite = not limited by any > threshold. The set of integers is a bit more complicated than a simple > mind may guess. There is no largest element. The set is not fixed. All > we know is that it has less than 10^100 elements. If I generate the first natural number in 1/2 second, the second in 1/4 second, and the nth natural number in 1/2^n seconds, I will clearly generate 10^100 before the first second has elapsed. karl m ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> threshold. The set of integers is a bit more complicated than a simple > mind may guess. There is no largest element. The set is not fixed. All > we know is that it has less than 10^100 elements. > If I generate the first natural number in 1/2 second, the second in 1/4 > second, and the nth natural number in 1/2^n seconds, I will clearly > generate 10^100 before the first second has elapsed. I know that it is hard to understand, but you cannot do anything unless your brain exists physically. Hence physical constraints apply to your thinking and to mathematics in general. You cannot generate more than 10^50 different elements in your brain --- far less. ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> threshold. The set of integers is a bit more complicated than a simple > mind may guess. There is no largest element. The set is not fixed. All > we know is that it has less than 10^100 elements. > If I generate the first natural number in 1/2 second, the second in 1/4 > second, and the nth natural number in 1/2^n seconds, I will clearly > generate 10^100 before the first second has elapsed. > I know that it is hard to understand, but you cannot do anything unless > your brain exists physically. Yes, over here we draw lists daily. Then we prioritize them. Then we begin to do each thing on the list with seven steps. What do you do over there? > Hence physical constraints apply to your > thinking and to mathematics in general. You cannot generate more than > 10^50 different elements in your brain --- far less. No. MATHEMATICS IN GENERAL admits no constraints. What's your story? malbrain ==== Subject: Re: Update: Objections to Cantor's Theory > I know that it is hard to understand, but you cannot do anything unless > your brain exists physically. Hence physical constraints apply to your > thinking and to mathematics in general. You cannot generate more than > 10^50 different elements in your brain --- far less. When I get to element 10^50 I just add one. There is no largest integers. There may be a largest integer which has a physical representation, but integers being abstract need not have physical representations. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory > If I generate the first natural number in 1/2 second, the second in 1/4 > second, and the nth natural number in 1/2^n seconds, I will clearly > generate 10^100 before the first second has elapsed. karl m True, but there will still be a largest natural on the output tape. Google an old thread titled No Set Contains Every Computable Natural. In that thread I give a proof that any computer that can perform an infinite number of operations in finite time can solve the Halting Problem. Since the Halting Problem is the diagonal argument in disguise, my proof solves Cantor's Diagonal argument. The diagonal argument (I would call it the missing number argument) can be made to produce a computable number. (The usual diagonal argument about real numbers doesn't produce a computable number). My definition of computable number means finite. Set theorist always want to invoke infinity even when it is overkill. Cantor's missing number argument doesn't have to perform an infinite number of operations to find the missing number. It can always find a missing number in a finite number of operations. Russell - 2 many 2 count ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> second, and the nth natural number in 1/2^n seconds, I will clearly > generate 10^100 before the first second has elapsed. karl m > True, but there will still be a largest natural on the output tape. > Google an old thread titled No Set Contains Every Computable Natural. Between sessions the parliamentarian maintains an active role with participants in developing their items for submission to chair for inclusion prior to the next meeting. POLEMICS WILL BE BASED ON OFFICIAL DOCUMENTS CONTAINED IN THE BIBLIOGRAPHY AND OFFICIAL AND SUPPLEMENTAL READING LISTS. You'll need a ruling from a higher body before we can bring forward material from that source. karl m ==== Subject: Re: Update: Objections to Cantor's Theory > No set is infinite, because the elements of a set must be distinguished > by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. Then you claim the set of integers is finite. If so, it has a large element n. What about n + 1. Apparently you are not bothered by flat out contradictions. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. > Then you claim the set of integers is finite. If so, it has a large > element n. What about n + 1. Apparently you are not bothered by flat out > contradictions. The values of integers are *potentially* infinite = not limited by any threshold. The set of integers is a bit more complicated than a simple mind may guess. There is no largest element. The set is not fixed. All we know is that it has less than 10^100 elements. See: Physical Constraints of Numbers, arxiv.org/abs/math.GM/0505649 ==== Subject: Re: Update: Objections to Cantor's Theory No set is infinite, because the elements of a set must be distinguished > by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. > Then you claim the set of integers is finite. If so, it has a large > element n. What about n + 1. Apparently you are not bothered by flat out > contradictions. > The values of integers are *potentially* infinite = not limited by any > threshold. The set of integers is a bit more complicated than a simple > mind may guess. There is no largest element. The set is not fixed. All > we know is that it has less than 10^100 elements. There is no such thing as WM's unfixed sets in any axiom system allowing sets that I am aware of. So whatever WM is talking about, it is not sets. Perhaps WM has an ultra-naive set theory that he wishes to promote as gospel, but he is not preaching to the choir here. ==== Subject: Re: Update: Objections to Cantor's Theory > The values of integers are *potentially* infinite = not limited by any > threshold. The set of integers is a bit more complicated than a > simple mind may guess... All > we know is that it has less than 10^100 elements. Consistency isn't a hugely important notion for you, is it? ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> <3kvabiF1062osU3@individual.net The values of integers are *potentially* infinite = not limited by any > threshold. The set of integers is a bit more complicated than a > > simple mind may guess... All > we know is that it has less than 10^100 elements. > Consistency isn't a hugely important notion for you, is it? The matter is not so easy as you might guess. The tools of set theory are copletely useless to understand things correctly. A set can have a finite number of elements, though it need not have a largest member. Sets are not that simple. They are not static things, in particular. ==== Subject: Re: Update: Objections to Cantor's Theory > The matter is not so easy as you might guess. The tools of set theory > are copletely useless to understand things correctly. A set can have a > finite number of elements, though it need not have a largest member. > Sets are not that simple. They are not static things, in particular. Produce a partially ordered finite set which does not have a maximal element with respect to the order. A set is a collection of object all satisfying some predicate. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> <3kvabiF1062osU3@individual.net> are copletely useless to understand things correctly. A set can have a > finite number of elements, though it need not have a largest member. > Sets are not that simple. They are not static things, in particular. > Produce a partially ordered finite set which does not have a maximal > element with respect to the order. The set of all natural numbers which can been individually named simultaneously within the whole universe. > A set is a collection of object all satisfying some predicate. Like the set of all humans which ever existed. This set has no last element (hitherto). It is continuously changing. ==== Subject: Re: Update: Objections to Cantor's Theory The matter is not so easy as you might guess. The tools of set theory > are copletely useless to understand things correctly. A set can have a > finite number of elements, though it need not have a largest member. > Sets are not that simple. They are not static things, in particular. > Produce a partially ordered finite set which does not have a maximal > element with respect to the order. > The set of all natural numbers which can been individually named > simultaneously within the whole universe. Not a set, at least in any mathematical sense, because its membership is ambiguous. For one thing, it appears to be be time dependent, which mathematical sets cannot be. ==== Subject: Re: Update: Objections to Cantor's Theory comcast.com> <3kvabiF1062osU3@individual.net> The matter is not so easy as you might guess. The tools of set theory > are copletely useless to understand things correctly. A set can have a > finite number of elements, though it need not have a largest member. > Sets are not that simple. They are not static things, in particular. > Produce a partially ordered finite set which does not have a maximal > element with respect to the order. > The set of all natural numbers which can been individually named > simultaneously within the whole universe. No. Our agreement with the numbers is that they would ORGANIZE THEMSELVES and that we would agree not to limit them in any way. This has already been proven in an EARLIER MOTION -- YOU CHOSE TO GO FIRST IN THE ENUMERATION. Do you say YES OR NO to the CURRENT MOTION ON THE FLOOR. karl m ==== Subject: Re: Update: Objections to Cantor's Theory > The values of integers are *potentially* infinite = not limited by any > threshold. The set of integers is a bit more complicated than a > > simple mind may guess... All > we know is that it has less than 10^100 elements. > Consistency isn't a hugely important notion for you, is it? > The matter is not so easy as you might guess. The tools of set theory > are copletely useless to understand things correctly. A set can have a > finite number of elements, though it need not have a largest member. > Sets are not that simple. They are not static things, in particular. In mathematical set theory, sets are static. In WMical set theory, nothing is static, including truth. ==== Subject: Re: Update: Objections to Cantor's Theory >The values of integers are *potentially* infinite = not limited by any >threshold. The set of integers is a bit more complicated than a > simple mind may guess... All >we know is that it has less than 10^100 elements. >>Consistency isn't a hugely important notion for you, is it? > The matter is not so easy as you might guess. The tools of set theory > are copletely useless to understand things correctly. Which things? And what do you propose to use instead? > A set can have a > finite number of elements, though it need not have a largest member. > Sets are not that simple. They are not static things, in particular. So saying that there is no upper limit to the number of integers, but that there are definitely less than 10^100 of them is made consistent by sets not being static. And I thought 'Jabberwocky' was weird... ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> <3kvabiF1062osU3@individual.net> <3kvjd6Fvsk8pU1@individual.net > A set can have a > finite number of elements, though it need not have a largest member. > Sets are not that simple. They are not static things, in particular. > So saying that there is no upper limit to the number of integers, > but that there are definitely less than 10^100 of them is > made consistent by sets not being static. You have understood what I said. At least you are able to repeat it correctly in your own words. That's more than most mathematicians accomplish. (Not because I was not clear enough, but because they are too much fixed to that stuff they were indoctrinated with, once upon a time.) ==== Subject: Re: Update: Objections to Cantor's Theory > > A set can have a > finite number of elements, though it need not have a largest member. > Sets are not that simple. They are not static things, in particular. > So saying that there is no upper limit to the number of integers, > but that there are definitely less than 10^100 of them is > made consistent by sets not being static. > You have understood what I said. At least you are able to repeat it > correctly in your own words. So WM says that there is no upper limit to the number of integers, and also says that 10^100 is an upper limit on the number of integers? > That's more than most mathematicians > accomplish. It is not that we do not understand what WM says, it is that we do not agree with it. Particularly when he says that something does not exist and that id does exist in virtually the same breath. > (Not because I was not clear enough, but because they are > too much fixed to that stuff they were indoctrinated with, once upon a > time.) We are too much indoctrinated with stuff called logic to agree to the simultaneous existence and non-existence of anything. WM should try it sometime. ==== Subject: Re: Update: Objections to Cantor's Theory >>So saying that there is no upper limit to the number of integers, >>but that there are definitely less than 10^100 of them is >>made consistent by sets not being static. > You have understood what I said. Not yet, I think. So, there are less than 10^100 integers. Is it also the case that all the integers are less than 10^100? (It seems unlikely that you're claiming this, because presumably (10^100)+1 is an integer. So there are less than 10^100, but there are gaps in the sequence once they get really big? ==== Subject: Re: Update: Objections to Cantor's Theory comcast.com> <3kvabiF1062osU3@individual.net> <3kvjd6Fvsk8pU1@individual.net> <3l4f02F10nc0oU2@individual.net>So saying that there is no upper limit to the number of integers, >>but that there are definitely less than 10^100 of them is >>made consistent by sets not being static. > You have understood what I said. > Not yet, I think. > So, there are less than 10^100 integers. Is it also > the case that all the integers are less than 10^100? > (It seems unlikely that you're claiming this, because > presumably (10^100)+1 is an integer. Of course it is. It is the same as with Cantor's list: You can enumerate any of the reals, but not *all* of them. In Cantor's arguing this result is due to a flaw (the assumption that all naturals exist), in my case it is clear from the facts that the (accessible part of the) universe is limited and will ever remain so. So there are > less than 10^100, but there are gaps in the sequence > once they get really big? Floor(Pi*10^10^100) cannot be realized. Nobody will ever know all of its digits, independently of whether in hexadecimal expansion some digits can easily be calculated. Have nice vacations. Further questions must be answered by yourself or your colleagues. Bye, Wolfgang Mueckenheim ==== Subject: Re: Update: Objections to Cantor's Theory >>So saying that there is no upper limit to the number of integers, >>but that there are definitely less than 10^100 of them is >>made consistent by sets not being static. > You have understood what I said. > Not yet, I think. > So, there are less than 10^100 integers. Is it also > the case that all the integers are less than 10^100? > (It seems unlikely that you're claiming this, because > presumably (10^100)+1 is an integer. > Of course it is. It is the same as with Cantor's list: You can > enumerate any of the reals, but not *all* of them. In Cantor's arguing > this result is due to a flaw (the assumption that all naturals exist) But WM is unable to give an example of one which does not. And WM us unable to give an example of any natural for which its successor is not a natural. In terms of physical existence, no mathematics exists at all and no naturals exist at all. So all mathematical existence is ideal, and WM's alleged physical limitations only affect the physical sciences, and are irrelevant for mathematics. Mathematics is no more physically dependent than chess is depencent of the existence of a particular physical chess set, or any physical chess set, as far as that goes. Chess putzers may require boards, but masters do not. WM may require a physical exemplar for his mathematics, but master mathematicians don't. ==== Subject: Re: Update: Objections to Cantor's Theory comcast.com> <3kvabiF1062osU3@individual.net> <3kvjd6Fvsk8pU1@individual.net > A set can have a > finite number of elements, though it need not have a largest member. > Sets are not that simple. They are not static things, in particular. > So saying that there is no upper limit to the number of integers, > but that there are definitely less than 10^100 of them is > made consistent by sets not being static. > You have understood what I said. At least you are able to repeat it > correctly in your own words. That's more than most mathematicians > accomplish. (Not because I was not clear enough, but because they are > too much fixed to that stuff they were indoctrinated with, once upon a > time.) There is an ENUMERATION being conducted for a motion brought by the chair. Is your position on the motion YES or No, or would you like to REQUEST that the motion be repeated? karl m ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> <3kvabiF1062osU3@individual.net The values of integers are *potentially* infinite = not limited by any > threshold. The set of integers is a bit more complicated than a > > simple mind may guess... All > we know is that it has less than 10^100 elements. > Consistency isn't a hugely important notion for you, is it? > The matter is not so easy as you might guess. The tools of set theory > are copletely useless to understand things correctly. A set can have a > finite number of elements, though it need not have a largest member. > Sets are not that simple. They are not static things, in particular. So, taking Barb's Java program, translated to C, as an illustration we can run for( i = 0; i++; ) print(i + i); and not expect the program to crash? karl m ==== Subject: Re: Update: Objections to Cantor's Theory > The values of integers are *potentially* infinite = not limited by any > threshold. The set of integers is a bit more complicated than a simple > mind may guess. There is no largest element. The set is not fixed. All > we know is that it has less than 10^100 elements. You are using the word infinite in a different sense. The set of integers is infinite in the sense that it is one to one correspondence with a proper subset of itself. Let N be the set of integers and E be the set of even intengers. The correspondence n <-> 2*n establishes the one to one mapping. The set of integers is infinite in this sense of the word infinite. There is no largest naturak number. If there were a largest say n, then n + 1 is not a natural, but this is false. If n is a nature number, then so is n + 1. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> Yes, there are some religious sects which, besides mathematicians, > believe in the infinite. > Is the set of integers finite or infinite? > No set is infinite, because the elements of a set must be distinguished > by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. > All other belief is but that. Yes, our belief is that our agreement to our AXIOMS will lead us forward from (the current) REALITY. karl m ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. Okay. So do you have an axiomatization for mathematics that doesn't include infinite sets? What's your logistic system, your primtive terms, and your axioms? MoeBlee ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. > Okay. So do you have an axiomatization for mathematics that doesn't > include infinite sets? What's your logistic system, your primtive > terms, and your axioms? I take what is possible. And i know that it is impossible to label more than 10^100 elements by different labels. I use and teach current mathematics, algebra and analysis, as usual because the physical constraints in this domain are not visible there. They become only visible in set theory. I hope hat some gifted mathematicians sometime will work out a consistent theory. ==== Subject: Re: Update: Objections to Cantor's Theory > No set is infinite, because the elements of a set must be distinguished > by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. > Okay. So do you have an axiomatization for mathematics that doesn't > include infinite sets? What's your logistic system, your primtive > terms, and your axioms? > I take what is possible. And try to impose the impossible. > And i know that it is impossible to label more > than 10^100 elements by different labels. The existence, at least in the world of the imagination where all mathematics takes place, of labels for everything is irrelevant. > I use and teach current mathematics, algebra and analysis, as usual > because the physical constraints in this domain are not visible there. But, according to a prior posting, WM requires his victims to learn his perverted catechism in spite of its irrelevance. And certainly WM's claim that his perversions are not visible in analysis requires a perversion of analysis. In every version of analysis that I am familiar with, there are irrational and transcendental numbers all over the place. > They become only visible in set theory. > I hope hat some gifted mathematicians sometime will work out a > consistent theory. They have. That WM does not like the theory does not render it inconsistent. ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. > Okay. So do you have an axiomatization for mathematics that doesn't > include infinite sets? What's your logistic system, your primtive > terms, and your axioms? > I take what is possible. And i know that it is impossible to label more > than 10^100 elements by different labels. > I use and teach current mathematics, algebra and analysis, as usual > because the physical constraints in this domain are not visible there. > They become only visible in set theory. > I hope hat some gifted mathematicians sometime will work out a > consistent theory. Waiting for Godot. MoeBlee ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu No set is infinite, because the elements of a set must be distinguished > by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. > Okay. So do you have an axiomatization for mathematics that doesn't > include infinite sets? What's your logistic system, your primtive > terms, and your axioms? > I take what is possible. And i know that it is impossible to label more > than 10^100 elements by different labels. The numbers have organized themselves into a SYSTEM where they name themselves. They demanded in our agreement that we not limit them. > I use and teach current mathematics, algebra and analysis, as usual > because the physical constraints in this domain are not visible there. > They become only visible in set theory. Which number failed to give its name when called upon? We have a provision in our agreement with them that they will repair any injury we find with them. > I hope hat some gifted mathematicians sometime will work out a > consistent theory. Oh, it was not a matter of any wise or clever man, just the course of human history side by side with the numbers. karl m ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> because the physical constraints in this domain are not visible there. > They become only visible in set theory. > Which number failed to give its name when called upon? We have a > provision in our agreement with them that they will repair any injury > we find with them. Each number < 10^100 is easily written down in, say, one minute. Generating it by a machine would be possible in far less than one second. But it is impossible to call all 10^100 of them. You should know the principle from Cantor's list. There is no real number which could not be included. But it is impossible to include all of them. The only difference is that Cantor's proof is humbug whereas my proof is evident from the state of our world. ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> because the physical constraints in this domain are not visible there. > They become only visible in set theory. > Which number failed to give its name when called upon? We have a > provision in our agreement with them that they will repair any injury > we find with them. > Each number < 10^100 is easily written down in, say, one minute. > Generating it by a machine would be possible in far less than one > second. But it is impossible to call all 10^100 of them. Nonsense. Our contract with the numbers has never failed. Which natural number refused to provide its name when called? > You should > know the principle from Cantor's list. I'm not familiar with Mr. Cantor's list. Can you reproduce it? > There is no real number which > could not be included. In our encoding there's a special NAN symbol for this STATE TRANSITION. > But it is impossible to include all of them. The > only difference is that Cantor's proof is humbug whereas my proof is > evident from the state of our world. SUPPLEMENTAL READING LIST does state of our world come from? malbrain ==== Subject: Re: Update: Objections to Cantor's Theory > I hope hat some gifted mathematicians sometime will work out a > consistent theory. Demonstrated how? Goedel has shown that in certain formal systems consistency is not provable. So far, no one, including you, has shown the various current set theories is inconsistant. Since consistency is not demonstrable except in systems with finite models, what you ask for cannot be provided. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> consistent theory. > Demonstrated how? Goedel has shown that in certain formal systems > consistency is not provable. So far, no one, including you, has shown > the various current set theories is inconsistant. Since consistency is > not demonstrable except in systems with finite models, what you ask for > cannot be provided. It has been shown that every consistent theory has a finite model. Set theory has not. Isn't that enough evidence? ==== Subject: Re: Update: Objections to Cantor's Theory I hope hat some gifted mathematicians sometime will work out a > consistent theory. > Demonstrated how? Goedel has shown that in certain formal systems > consistency is not provable. So far, no one, including you, has shown > the various current set theories is inconsistant. Since consistency is > not demonstrable except in systems with finite models, what you ask for > cannot be provided. > It has been shown that every consistent theory has a finite model. References? Goedel's completeness theorem says that a theory has a model if and only if it is consistent, but I have not found any reference to what WM claims. ==== Subject: Re: Update: Objections to Cantor's Theory comcast.com> consistent theory. Demonstrated how? Goedel has shown that in certain formal systems > consistency is not provable. So far, no one, including you, has shown > the various current set theories is inconsistant. Since consistency is > not demonstrable except in systems with finite models, what you ask for > cannot be provided. > It has been shown that every consistent theory has a finite model. > References? > Goedel's completeness theorem says that a theory has a model if and only > if it is consistent, but I have not found any reference to what WM > claims. Skolem extended work by L.9awenheim (published in 1915) to give the L.9awenheim-Skolem theorem, which he published in 1920. It states that if a theory within first-order predicate calculus has a model then it has a countable model. His 1920 proof of this result used the axiom of choice, but later in 1922 and 1928 he gave proofs using K.9anig's lemma (due to Julius K.9anig) which do not require the axiom of choice. He made refinements to Zermelo's axiomatic set theory, publishing work in 1922 and 1929. The first was the published version of the lecture Einige Bemerkungen zur axiomatischen Begr.9fndung der Mengenlehre which he gave in 1922 at the 5th Scandinavian Mathematics Congress. Here he applied the L.9awenheim-Skolem theorem to show what became known as Skolem's paradox: If the Zermelo's axiomatic system for set theory is consistent then it must be satisfiable within a countable domain. Skolem is commonly portrayed as arguing that certain otherwise well understood concepts are suspect simply because they cannot be characterized in a first-order language; in particular that, since all first-order formalizations of set theory (if consistent) have countable models, the concept of uncountability is flawed. ... Skolem's position is more solid than that. I see Skolem as arguing that all the evidence that has been given for the existence of uncountable sets is inconclusive ... Google for Skolem. Have nice vacations. I hope to have some. Bye, WM ==== Subject: Re: Update: Objections to Cantor's Theory > I hope hat some gifted mathematicians sometime will work out a > consistent theory. Demonstrated how? Goedel has shown that in certain formal systems > consistency is not provable. So far, no one, including you, has shown > the various current set theories is inconsistant. Since consistency is > not demonstrable except in systems with finite models, what you ask for > cannot be provided. It has been shown that every consistent theory has a finite model. > References? > Goedel's completeness theorem says that a theory has a model if and only > if it is consistent, but I have not found any reference to what WM > claims. > Skolem extended work by L.9awenheim (published in 1915) to give the > L.9awenheim-Skolem theorem, which he published in 1920. It states that > if a theory within first-order predicate calculus has a model then it > has a countable model. Other than in WM's post-Skolem claims, when does countable imply finite? If Skolem and Loewenheim say countable, they must be presumed to be including countably infinite. AS usual WM is claiming much more that he is able to provide. ==== Subject: Re: Update: Objections to Cantor's Theory >It has been shown that every consistent theory has a finite model. >>Goedel's completeness theorem says that a theory has a model if and only >>if it is consistent, but I have not found any reference to what WM >>claims. > Skolem extended work by L.9awenheim (published in 1915) to give the > L.9awenheim-Skolem theorem, which he published in 1920. It states that > if a theory within first-order predicate calculus has a model then it > has a countable model. Except perhaps in your strange parallel universe, that is *not* a finite model, but a countably infinite model. (And, indeed, there is a model of any given cardinality.) ==== Subject: Re: Update: Objections to Cantor's Theory > Except perhaps in your strange parallel universe, that is > *not* a finite model, but a countably infinite model. > (And, indeed, there is a model of any given cardinality.) Sigh. I meant, of course, 'there is a model of any given infinite cardinality'. ==== Subject: Re: Update: Objections to Cantor's Theory > Goedel's completeness theorem says that a theory has a model if and only > if it is consistent, but I have not found any reference to what WM > claims. You are surely aware that there are consistent theories that have no finite models. Why this obsession with Mueckenherz's peculiar thought processes? ==== Subject: Re: Update: Objections to Cantor's Theory comcast.com> if it is consistent, but I have not found any reference to what WM > claims. > You are surely aware that there are consistent theories that have no > finite models. Are we in logic here without any professional? Skolem extended work by L.9awenheim (published in 1915) to give the L.9awenheim-Skolem theorem, which he published in 1920. It states that if a theory within first-order predicate calculus has a model then it has a countable model. His 1920 proof of this result used the axiom of choice, but later in 1922 and 1928 he gave proofs using K.9anig's lemma (due to Julius K.9anig) which do not require the axiom of choice. He made refinements to Zermelo's axiomatic set theory, publishing work in 1922 and 1929. The first was the published version of the lecture Einige Bemerkungen zur axiomatischen Begr.9fndung der Mengenlehre which he gave in 1922 at the 5th Scandinavian Mathematics Congress. Here he applied the L.9awenheim-Skolem theorem to show what became known as Skolem's paradox: If the Zermelo's axiomatic system for set theory is consistent then it must be satisfiable within a countable domain. Skolem is commonly portrayed as arguing that certain otherwise well understood concepts are suspect simply because they cannot be characterized in a first-order language; in particular that, since all first-order formalizations of set theory (if consistent) have countable models, the concept of uncountability is flawed. ... Skolem's position is more solid than that. I see Skolem as arguing that all the evidence that has been given for the existence of uncountable sets is inconclusive. Google for Skolem. Have nice vacations. I hope to have some. Bye, WM ==== Subject: Re: Update: Objections to Cantor's Theory > Goedel's completeness theorem says that a theory has a model if and only > if it is consistent, but I have not found any reference to what WM > claims. > You are surely aware that there are consistent theories that have no > finite models. > Are we in logic here without any professional? > Skolem extended work by L.9awenheim (published in 1915) to give the > L.9awenheim-Skolem theorem, which he published in 1920. It states that > if a theory within first-order predicate calculus has a model then it > has a countable model. Countable does not mean finite! WM's theory requires the set of naturals to be finite. Skolem's theorem requires it to be infinite. > His 1920 proof of this result used the axiom of > choice, but later in 1922 and 1928 he gave proofs using K.9anig's lemma > (due to Julius K.9anig) which do not require the axiom of choice. > He made refinements to Zermelo's axiomatic set theory, publishing work > in 1922 and 1929. The first was the published version of the lecture > Einige Bemerkungen zur axiomatischen Begr.9fndung der Mengenlehre which > he gave in 1922 at the 5th Scandinavian Mathematics Congress. Here he > applied the L.9awenheim-Skolem theorem to show what became known as > Skolem's paradox: If the Zermelo's axiomatic system for set theory is > consistent then it must be satisfiable within a countable domain. Countable does not mean finite! WM's theory requires the set of naturals to be finite. Skolem's theorem requires it to be infinite. > Skolem is commonly portrayed as arguing that certain otherwise well > understood concepts are suspect simply because they cannot be > characterized in a first-order language; in particular that, since all > first-order formalizations of set theory (if consistent) have countable > models, the concept of uncountability is flawed. ... Skolem's position > is more solid than that. I see Skolem as arguing that all the evidence > that has been given for the existence of uncountable sets is > inconclusive. But AFAIK there is no first order model of set theory which allows all of natural umber arithmetic. The way I understand it, while one can get addition of naturals, one cannot even get multiplication, if forced to stick with first order logic. So as long as one does not need multiplication, one does not need uncountable sets. What does that do to physics? ==== Subject: Re: Update: Objections to Cantor's Theory > Goedel's completeness theorem says that a theory has a model if and only > if it is consistent, but I have not found any reference to what WM > claims. > You are surely aware that there are consistent theories that have no > finite models. Why this obsession with Mueckenherz's peculiar thought > processes? I am aware of it, but WM is apparently not, and is trying to foist another of his many falsehoods on us. I am fascinated by WM's powers of self-deception. At least WM seems too earnest to be merely trolling, though I could be wrong about that. ==== Subject: Re: Update: Objections to Cantor's Theory > You are surely aware that there are consistent theories that have no > finite models. Why this obsession with Mueckenherz's peculiar thought > processes? Trying to understand the point of view of somebody that peculiar has its own insidious charm. At the very least, it can be entertaining to watch the contortions they're willing to go through when questioned moderately closely. I know, it's undignified. ==== Subject: Re: Update: Objections to Cantor's Theory > It has been shown that every consistent theory has a finite model. Damn, I missed that one. Where? > Set > theory has not. Isn't that enough evidence? I'd like to see the proof that 'every consistent theory has a finite model'. ==== Subject: Re: Update: Objections to Cantor's Theory > It has been shown that every consistent theory has a finite model. Set > theory has not. Isn't that enough evidence? The converse of that is true. Any system with a finite model is consistent. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory >> It has been shown that every consistent theory has a finite model. Set >> theory has not. Isn't that enough evidence? > The converse of that is true. Any system with a finite model is consistent. I can believe that, since any system with a model (finite or not) is consistent. ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> I hope hat some gifted mathematicians sometime will work out a > consistent theory. > Demonstrated how? Goedel has shown that in certain formal systems > consistency is not provable. So far, no one, including you, has shown > the various current set theories is inconsistant. Since consistency is > not demonstrable except in systems with finite models, what you ask for > cannot be provided. > It has been shown that every consistent theory has a finite model. Set > theory has not. Isn't that enough evidence? Once the chair has called for the ENUMERATION, the first step is the PROOF THAT A QUORUM of MEMBERS took part in it. If this PROOF FAILS, the motion is invalidated for lack of a SECOND. So, no, it is not a question of evidence -- it is a question of YES or NO to the MOTION. karl m ==== Subject: Re: Update: Objections to Cantor's Theory !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > No set is infinite, because the elements of a set must be distinguished > by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. >> Okay. So do you have an axiomatization for mathematics that doesn't >> include infinite sets? What's your logistic system, your primtive >> terms, and your axioms? > I take what is possible. And i know that it is impossible to label > more than 10^100 elements by different labels. I'd use 100-digit integers for that. So which 100-digit integer is not possible to use a label, according to you? > I use and teach current mathematics, algebra and analysis, Which is quite a travesty and shows the low standards of the Augsburger Fachhochschule. > I hope hat some gifted mathematicians sometime will work out a > consistent theory. Happened centuries ago. You just don't get it, but given your demonstrated inability to grasp fundamental concepts like quantifiers, that is nothing that can be helped. -- ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> Yes, there are some religious sects which, besides mathematicians, > believe in the infinite. > Is the set of integers finite or infinite? > No set is infinite, because the elements of a set must be distinguished > by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. Yes, that's a contradiction, but it's a contradiction of PHYSICS, not MATHEMATICS. Time is not a contradiction in mathematics. karl m ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.edu> Yes, there are some religious sects which, besides mathematicians, > believe in the infinite. > Is the set of integers finite or infinite? > No set is infinite, because the elements of a set must be distinguished > by at least one property. For that sake one needs at least one part of > reality. But the part(icle)s of reality are finite. If there's one thing that the set of integers needs from reality, it's ONE. Each integer differes from the next by ONE. You even take it for granted yourself. ZERO is an axiom. The system gives us ONE. karl m ==== Subject: Re: Update: Objections to Cantor's Theory >>Then pray tell what the 'true' definition of more is. >>What does it mean for a set to have more elements than >>another set? > Apparently it depends on one's profession but since mathematicians > make up a minority of all people, its silly to think that any meaning > they make up should be accepted by all people. Undoubtedly when > non-mathematicians are called cranks because they disagree with > arcane, and admittedly (by mathematicians) informal meanings of > common words, then mathematicians are the people that are wrong. You still have not said what it means for a set to have more elements than another set. It should be an easy question. However I suspect you are just trolling. Sets are mathematical objects, and I think it makes perfect sense for mathematicians to decides what words mean when describing sets Stephen ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> Then pray tell what the 'true' definition of more is. >>What does it mean for a set to have more elements than >>another set? > Apparently it depends on one's profession but since mathematicians > make up a minority of all people, its silly to think that any meaning > they make up should be accepted by all people. Undoubtedly when > non-mathematicians are called cranks because they disagree with > arcane, and admittedly (by mathematicians) informal meanings of > common words, then mathematicians are the people that are wrong. > You still have not said what it means for a set to have > more elements than another set. It should be an easy > question. However I suspect you are just trolling. Cardinality is that general property which by means of our general capability of thinking can be attached to a set if the order of its elements and their properties are not taken into account. (Cantor, Collected works p. 282, my translation) This capability of thinking may be applied in the following form: Given two infinite sets A and B with elements a e A and b e B. The union of these sets does exist. If the elements can be put into an order < (not necessarily a well-order) such that in this order 1) there are all elements a e A and b e B 2) there are never two elements b,b' e B without an element a e A between them with respect to < then the cardinality Card(B) of B is not larger than the cardinality Card(A) of A: Card(B) =< Card(A). ==== Subject: Re: Update: Objections to Cantor's Theory >>Then pray tell what the 'true' definition of more is. >>What does it mean for a set to have more elements than >>another set? > Apparently it depends on one's profession but since mathematicians > make up a minority of all people, its silly to think that any meaning > they make up should be accepted by all people. Undoubtedly when > non-mathematicians are called cranks because they disagree with > arcane, and admittedly (by mathematicians) informal meanings of > common words, then mathematicians are the people that are wrong. > You still have not said what it means for a set to have > more elements than another set. It should be an easy > question. However I suspect you are just trolling. > Cardinality is that general property which by means of our general > capability of thinking can be attached to a set if the order of its > elements and their properties are not taken into account. (Cantor, > Collected works p. 282, my translation) > This capability of thinking may be applied in the following form: > Given two infinite sets A and B with elements a e A and b e B. The > union of these sets does exist. If the elements can be put into an > order < (not necessarily a well-order) such that in this order > 1) there are all elements a e A and b e B > 2) there are never two elements b,b' e B without an element a e A > between them with respect to < > then the cardinality Card(B) of B is not larger than the cardinality > Card(A) of A: > Card(B) =< Card(A). This axiom, and it is no more that an assumption made without factual support, is not supported by anything except WM's faith in it. Wm cannot prove it except by assuming either it or some equivalent statement. Since WM elsewhere objects to allowing such assumptions, he violates his own principles to propose it. ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> more elements than another set. It should be an easy > question. However I suspect you are just trolling. > Cardinality is that general property which by means of our general > capability of thinking can be attached to a set if the order of its > elements and their properties are not taken into account. (Cantor, > Collected works p. 282, my translation) > This capability of thinking may be applied in the following form: > Given two infinite sets A and B with elements a e A and b e B. The > union of these sets does exist. If the elements can be put into an > order < (not necessarily a well-order) such that... Huh? You just quoted Cantor above saying that cardinality refers to a property of a set where the order is /not/ taken into account. So why are you talking about order? > in this order > 1) there are all elements a e A and b e B > 2) there are never two elements b,b' e B without an element a e A > between them with respect to < Yes, those are just your premises regarding rationals/irrationals, restated more generally; this part of your argument doesn't need any more clarification. > then the cardinality Card(B) of B is not larger than the cardinality > Card(A) of A: > Card(B) =< Card(A). This does not follow; /why/ ... then ...? If you teach mathematics, surely you must see this requires proof, not just assertion. ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> more elements than another set. It should be an easy > question. However I suspect you are just trolling. > Cardinality is that general property which by means of our general > capability of thinking can be attached to a set if the order of its > elements and their properties are not taken into account. (Cantor, > Collected works p. 282, my translation) > This capability of thinking may be applied in the following form: > Given two infinite sets A and B with elements a e A and b e B. The > union of these sets does exist. If the elements can be put into an > order < (not necessarily a well-order) such that... > Huh? You just quoted Cantor above saying that cardinality refers to a > property of a set where the order is /not/ taken into account. So why > are you talking about order? Cardinality does not depend on order but it cannot be measured and sets cannot be compared unless you have some order. In most cases even a well-ordered set is required. > This does not follow; /why/ ... then ...? If you teach mathematics, > surely you must see this requires proof, not just assertion. It is obvious. ==== Subject: Re: Update: Objections to Cantor's Theory > You still have not said what it means for a set to have > more elements than another set. It should be an easy > question. However I suspect you are just trolling. Cardinality is that general property which by means of our general > capability of thinking can be attached to a set if the order of its > elements and their properties are not taken into account. (Cantor, > Collected works p. 282, my translation) This capability of thinking may be applied in the following form: > Given two infinite sets A and B with elements a e A and b e B. The > union of these sets does exist. If the elements can be put into an > order < (not necessarily a well-order) such that... > Huh? You just quoted Cantor above saying that cardinality refers to a > property of a set where the order is /not/ taken into account. So why > are you talking about order? > Cardinality does not depend on order but it cannot be measured and sets > cannot be compared unless you have some order. In most cases even a > well-ordered set is required. If AC in assumed, as in ZFC, a well ordering always can be created. But no a priori ordering is required. > This does not follow; /why/ ... then ...? If you teach mathematics, > surely you must see this requires proof, not just assertion. > It is obvious. Such declarations do not constitute mathematically valid proofs, and, in fact, no a priori ordering is required in ZFC. And in general, while the ability to well order every set is certainly sufficient, WM has not shown it to be necessary. Certainly the sets of nodes in any two maximal binary trees can be put into bijection without any attempt to impose a total order on those nodes, so in this case the tree structure, which is nothing like a (total) ordering of the nodes, is quite sufficient. ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> cannot be compared unless you have some order. In most cases even a > well-ordered set is required. > If AC in assumed, as in ZFC, a well ordering always can be created. But > no a priori ordering is required. > This does not follow; /why/ ... then ...? If you teach mathematics, > surely you must see this requires proof, not just assertion. > It is obvious. > Such declarations do not constitute mathematically valid proofs, and, in > fact, no a priori ordering is required in ZFC. > And in general, while the ability to well order every set is certainly > sufficient, WM has not shown it to be necessary. Zermelo like Cantor recognized that it was necessary, in order to attach an aleph to every set. Therefore AC was created. Therefore I need not show it. But you should know it. > Certainly the sets of nodes in any two maximal binary trees can be put > into bijection without any attempt to impose a total order on those > nodes, so in this case the tree structure, which is nothing like a > (total) ordering of the nodes, is quite sufficient. The tree structure is an order with no doubt. And I spoke of some order in my original contribution (see above). ==== Subject: Re: Update: Objections to Cantor's Theory > Cardinality does not depend on order but it cannot be measured and sets > cannot be compared unless you have some order. In most cases even a > well-ordered set is required. > If AC in assumed, as in ZFC, a well ordering always can be created. But > no a priori ordering is required. This does not follow; /why/ ... then ...? If you teach mathematics, > surely you must see this requires proof, not just assertion. It is obvious. > Such declarations do not constitute mathematically valid proofs, and, in > fact, no a priori ordering is required in ZFC. > And in general, while the ability to well order every set is certainly > sufficient, WM has not shown it to be necessary. > Zermelo like Cantor recognized that it was necessary, in order to > attach an aleph to every set. Not every set needs an aleph attached to it in order to be shown of the same cardinality as some other set if there can be a bijection without an ordering > Therefore AC was created. Therefore I > need not show it. I have already said that, with AC, it is possible to well-order any set, WM has not shown that it is NECESSARY in order to compare two sets, to well order them, as he has claimed. > Certainly the sets of nodes in any two maximal binary trees can be put > into bijection without any attempt to impose a total order on those > nodes, so in this case the tree structure, which is nothing like a > (total) ordering of the nodes, is quite sufficient. > The tree structure is an order with no doubt. A tree structure is, at most, a partial order, not a total order. WM's relative cardinality requires a total order, so requires the axiom of choice. Cantor cardinality may apply even in the absence of a total ordering, so it may apply in the absence of AC. If WM wants to argue mathematics, he should learn some. ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> more elements than another set. It should be an easy > question. However I suspect you are just trolling. Cardinality is that general property which by means of our general > capability of thinking can be attached to a set if the order of its > elements and their properties are not taken into account. (Cantor, > Collected works p. 282, my translation) This capability of thinking may be applied in the following form: > Given two infinite sets A and B with elements a e A and b e B. The > union of these sets does exist. If the elements can be put into an > order < (not necessarily a well-order) such that... > Huh? You just quoted Cantor above saying that cardinality refers to a > property of a set where the order is /not/ taken into account. So why > are you talking about order? > Cardinality does not depend on order ... OK. > but it cannot be measured ... Why not? > and sets > cannot be compared unless you have some order. Erm? Surely we don't /need/ an order to compare the set {red, green, blue} with {robin, grackle, bunting}. And, if you accept them as sets, N and R (in their usual forms) can be compared in many ways without reference to order. For example, N is a proper subset of R. That seems to contradict your statement that infinite sets cannot be compared. On the other hand, if you don't consider N a set, then N and R don't even exist as sets, and it's absurd to talk about them as if they were sets. > In most cases even a > well-ordered set is required. Well, sure, if every set is finite, every set can be well-ordered. > This does not follow; /why/ ... then ...? If you teach mathematics, > surely you must see this requires proof, not just assertion. > It is obvious. It's not obvious to me; instead it seems obviously a non-sequitur. Does that mean you don't have a proof beyond your assertion, or that you choose not to present one? ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> surely you must see this requires proof, not just assertion. > It is obvious. > It's not obvious to me; instead it seems obviously a non-sequitur. > Does that mean you don't have a proof beyond your assertion, or that > you choose not to present one? It is, for finite sets, as obvious as the result of a bijection (if we add 1 element). For infinite sets we cannot prove anything (because those sets do not exist --- but that is not the point) but only extrapolate from the finite domain. There is no reason why we should prefer one of the two methods available. Cantor's bijection, because it is the olderone? Mathematicians, I know, are very conservative people. ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> surely you must see this requires proof, not just assertion. It is obvious. It's not obvious to me; instead it seems obviously a non-sequitur. > Does that mean you don't have a proof beyond your assertion, or that > you choose not to present one? > It is, for finite sets, as obvious as the result of a bijection (if we > add 1 element). For infinite sets we cannot prove anything (because > those sets do not exist --- but that is not the point) but only > extrapolate from the finite domain. There is no reason why we should > prefer one of the two methods available. Cantor's bijection, because it > is the olderone? Mathematicians, I know, are very conservative people. While it is a true saying that To a hammer, everything looks like a nail, I nonetheless prefer the tried and true method of using a hammer for nails, instead of a screwdriver. Not that I haven't used a screwdriver in a pinch... ==== Subject: Re: Update: Objections to Cantor's Theory > This does not follow; /why/ ... then ...? If you teach mathematics, > surely you must see this requires proof, not just assertion. It is obvious. It's not obvious to me; instead it seems obviously a non-sequitur. > Does that mean you don't have a proof beyond your assertion, or that > you choose not to present one? > It is, for finite sets, as obvious as the result of a bijection (if we > add 1 element). Much less obvious, particularly since the mechanism for showing two sets are of the same size is so screwed up. > For infinite sets we cannot prove anything (because > those sets do not exist --- but that is not the point) but only > extrapolate from the finite domain. There is no reason why we should > prefer one of the two methods available. If nothing else, I would prefer the method proposed by a mathematician and vetted by thousands of mathematicians since, to one proposed by an anti-mathematician and not even successfully vetted by that anti-mathematician. ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> blue} with {robin, grackle, bunting}. In order to show that every cardinality is an aleph we need the proof that every set can be well-ordered. Zermelo created AC just for that reason. Your above given sets are well ordered. Look at them. You will certainly see it. (You do not need the given order, but you need some order.) > And, if you accept them as sets, N and R (in their usual forms) can be > compared in many ways without reference to order. In their usual from, they are ordered (not well-ordered). Without any order there is no chance to compare sets. > For example, N is a > proper subset of R. That seems to contradict your statement that > infinite sets cannot be compared. > On the other hand, if you don't consider N a set, then N and R don't > even exist as sets, and it's absurd to talk about them as if they were > sets. > In most cases even a > well-ordered set is required. > Well, sure, if every set is finite, every set can be well-ordered. Every set, finite or infinite, can be well-ordered according to AC. For that sake this axiom was created. ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> blue} with {robin, grackle, bunting}. > In order to show that every cardinality is an aleph we need the proof > that every set can be well-ordered. Zermelo created AC just for that > reason. Your above given sets are well ordered. Look at them. You will > certainly see it. > (You do not need the given order, but you need some order.) > And, if you accept them as sets, N and R (in their usual forms) can be > compared in many ways without reference to order. > In their usual from, they are ordered (not well-ordered). Without any > order there is no chance to compare sets. Given any natural m, let S be the set of all abelian simple groups of order at most m. Let P be the set of all cyclic groups of order p, p prime and p<=m. Then S = P. Any set surely has the same size as itself; so |S| = |P|. I don't /need/ to place an order on S or P to compare them or even equate them; it follows from the usual group axioms and PA. > For example, N is a > proper subset of R. That seems to contradict your statement that > infinite sets cannot be compared. > On the other hand, if you don't consider N a set, then N and R don't > even exist as sets, and it's absurd to talk about them as if they were > sets. > In most cases even a > well-ordered set is required. Well, sure, if every set is finite, every set can be well-ordered. > Every set, finite or infinite, can be well-ordered according to AC. For > that sake this axiom was created. Believe it or not, just as you don't appear to accept AoI, some people don't accept AC. ==== Subject: Re: Update: Objections to Cantor's Theory > Erm? Surely we don't /need/ an order to compare the set {red, green, > blue} with {robin, grackle, bunting}. > In order to show that every cardinality is an aleph we need the proof > that every set can be well-ordered. Presumes two things not in evidence. (1) finite sets do not have an aleph cardinality. (2) it is not known to be necessary to find the aleph of a set, or to well order a set, to compare it to another set. > In their usual from, they are ordered (not well-ordered). Without any > order there is no chance to compare sets. Even if some sort of ordering were needed to compare, the ordering may be only a partial ordering, as with the example of maximal binary trees. But so what? In ZFC, for example, one has an axiom of choice, so that Cantor Cardinality works fine. If WM wants to work without a net (axiom system) fine, but mathematicians have learned the hard way that such recklessness is potentially disastrous. ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> more elements than another set. It should be an easy > question. However I suspect you are just trolling. Cardinality is that general property which by means of our general > capability of thinking can be attached to a set if the order of its > elements and their properties are not taken into account. (Cantor, > Collected works p. 282, my translation) This capability of thinking may be applied in the following form: > Given two infinite sets A and B with elements a e A and b e B. The > union of these sets does exist. If the elements can be put into an > order < (not necessarily a well-order) such that... > Huh? You just quoted Cantor above saying that cardinality refers to a > property of a set where the order is /not/ taken into account. So why > are you talking about order? > Cardinality does not depend on order but it cannot be measured and sets > cannot be compared unless you have some order. In most cases even a > well-ordered set is required. No, your notiton of cardinality is inconsistent with our agreement with the numbers. They organize themselves to respond when called upon, and we agreed not to limit them. That includes imposing, apriori, any order. Check with your GENERATING CONTRACT. karl m karl m ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> cannot be compared unless you have some order. In most cases even a > well-ordered set is required. > No, your notiton of cardinality is inconsistent with our agreement with > the numbers. They organize themselves to respond when called upon, and > we agreed not to limit them. That includes imposing, apriori, any > order. Check with your GENERATING CONTRACT. karl m The existence does not depend on an agreement. And if considered under physical aspects, we would find that there are no irrational numbers at all. But even admitting the existence of irrational numbers we find that two irrational numbers are not different, i.e., they are only one number, unless there exists at least one rational number between them. You can also say: Every irrational number needs at least one rational number in order to separate it from the smaller irrational numbers. This proves for anybody who is willing (and able) to think, that there cannot exist less rational numbers than irrational numbers. Therefore Cantor's proofs are either wrong or, at best, make set theory inconsistent. ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> cannot be compared unless you have some order. In most cases even a > well-ordered set is required. > No, your notiton of cardinality is inconsistent with our agreement with > the numbers. They organize themselves to respond when called upon, and > we agreed not to limit them. That includes imposing, apriori, any > order. Check with your GENERATING CONTRACT. karl m > The existence does not depend on an agreement. And if considered under > physical aspects, we would find that there are no irrational numbers at > all. Good lord, do you teach physics as well as mathematics!? ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> cannot be compared unless you have some order. In most cases even a > well-ordered set is required. > No, your notiton of cardinality is inconsistent with our agreement with > the numbers. They organize themselves to respond when called upon, and > we agreed not to limit them. That includes imposing, apriori, any > order. Check with your GENERATING CONTRACT. karl m > The existence does not depend on an agreement. Ok, so they can be COLLECTED by anyone with suitable apparatus. > And if considered under > physical aspects, Whoa. Are you sure you don't want alt.philosophy? > we would find that there are no irrational numbers at > all. Yes. If you force a large enough army, where they cannot see between the two opposite corners, to march from point A to point C by issuing orders at each and every step, you'll send them spiraling outwards. We cannot afford this. malbrain ==== Subject: Re: Update: Objections to Cantor's Theory !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > Cardinality does not depend on order but it cannot be measured and sets > cannot be compared unless you have some order. In most cases even a > well-ordered set is required. >> No, your notiton of cardinality is inconsistent with our agreement with >> the numbers. They organize themselves to respond when called upon, and >> we agreed not to limit them. That includes imposing, apriori, any >> order. Check with your GENERATING CONTRACT. karl m > The existence does not depend on an agreement. Quite so: not even on an agreement with physics. > And if considered under physical aspects, we would find that there > are no irrational numbers at all. But even admitting the existence > of irrational numbers we find that two irrational numbers are not > different, i.e., they are only one number, unless there exists at > least one rational number between them. Quite so. But if there exists _one_ rational number between them, then a countably infinite number of rational numbers exist between them, and an uncountably infinite number of irrational numbers. You don't get gaps with single numbers in them. > You can also say: Every irrational number needs at least one > rational number in order to separate it from the smaller irrational > numbers. This proves for anybody who is willing (and able) to > think, that there cannot exist less rational numbers than irrational > numbers. Therefore Cantor's proofs are either wrong or, at best, > make set theory inconsistent. But there is no alteration of numbers involved here at all. If you have a non-empty open interval, it always contains countably infinitely many rational and uncountably infinitely many irrational numbers. Not just a single one of each. -- ==== Subject: Re: Update: Objections to Cantor's Theory > Cardinality does not depend on order but it cannot be measured and sets > cannot be compared unless you have some order. In most cases even a > well-ordered set is required. > No, your notiton of cardinality is inconsistent with our agreement with > the numbers. They organize themselves to respond when called upon, and > we agreed not to limit them. That includes imposing, apriori, any > order. Check with your GENERATING CONTRACT. karl m > The existence does not depend on an agreement. The existence of agreement on what natural numbers are depends on argeement. If, as seems to be the case, WM find no agreement on his version, then his version has no general existence. WM is still, of course, quite free to play with them in the privacy of his home, but should not bring them into the light of public scrutiny. ==== Subject: Re: Update: Objections to Cantor's Theory > The existence does not depend on an agreement. And if considered under > physical aspects, we would find that there are no irrational numbers at > all. In the crude physcial world there are no integers at all. Not a one. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory >> The existence does not depend on an agreement. And if considered under >> physical aspects, we would find that there are no irrational numbers at >> all. > In the crude physcial world there are no integers at all. Not a one. There are no sounds. The sky isn't blue. Today doesn't exist. Ham doesn't have a flavor. Somehow ostrich scientists feel they are the only ones that live in a world of perceptions and ideas. ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> The existence does not depend on an agreement. And if considered under >> physical aspects, we would find that there are no irrational numbers at >> all. > In the crude physcial world there are no integers at all. Not a one. > There are no sounds. The sky isn't blue. Today doesn't exist. Ham > doesn't have a flavor. Somehow ostrich scientists feel they are the > only ones that live in a world of perceptions and ideas. The chair has asked for YOUR POSITION TO BE CLEARLY STATED ON THE QUESTION: Is that the considered judgement of this NG's ostrich-in-chief? karl m ==== Subject: Re: Update: Objections to Cantor's Theory su.edu> I don't really care what you think. I suspect you just like to call people names like crank and troll. ==== Subject: Re: Update: Objections to Cantor's Theory ws.msu.eduWhat does 'more reals than naturals' mean? You, like many others, >seem to be assuming there is some 'true' definition of more. > Most english speaking non-mathemeticians understand the meaning > of the words they use, so yes, we admit to such knowledge. When concepts arise in mathematics which are not part of ordinary experience or ordinary language, they must be given new names. There is no common English meaning for more as applied to infinite collections. The common sense of infinite, if there is one, is that it represents the ultimate in size, that there is nothing bigger. In mathematics, however, it was discovered that infinite sets can be classified by the ability to draw bijections between them. So it became useful to give a name to this property that any mapping from set A to set B must be incomplete, must be missing elements of B. The phrase larger cardinality was coined to describe this relationship. > If you mean something else by 'more', you have to define precisely > what you mean by it. > Do mathemeticians need help with simple word meanings? Only when dealing with new concepts not part of ordinary experience. The concept of comparing infinitely large things is not something that occurs in ordinary language. > Describe what you mean by if mathematically. > Describe what you mean by you mathematically. > Describe what you mean by accept mathematically. > Describe what you mean by that mathematically. > Describe what you mean by the mathematically. > Describe what you mean by are mathematically. These are all used in their ordinary sense. There is no ordinary sense of relative size of infinite sets. - Randy ==== Subject: Re: Update: Objections to Cantor's Theory !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > What does 'more reals than naturals' mean? You, like many others, > seem to be assuming there is some 'true' definition of more. In > the context of set theory 'more' is not well defined, but > informally many people will say B has more elements than A > if there does not exist a surjective mapping from A to B. AND there exists a surjective mapping from B to A. -- ==== Subject: Re: Update: Objections to Cantor's Theory >>What does 'more reals than naturals' mean? You, like many others, >>seem to be assuming there is some 'true' definition of more. In >>the context of set theory 'more' is not well defined, but >>informally many people will say B has more elements than A >>if there does not exist a surjective mapping from A to B. > AND there exists a surjective mapping from B to A. It's a theorem in ZFC that given any two sets A and B, there's at least one of a surjection from A to B and a surjection from B to A, isn't it? ==== Subject: Re: Update: Objections to Cantor's Theory > So, is it that you're unaware of all the attempts to provide > a fundamental model of space-time based on discrete models > (causal sets, spin networks etc), or you regard all that as > a fundamentally pointless research programme? When such an approach produces better result that those based on continuous compact domains, you will let me know? There's a good fellow. So far Wolfram's work using tessilated automota is a bust. Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory >> So, is it that you're unaware of all the attempts to provide >> a fundamental model of space-time based on discrete models >> (causal sets, spin networks etc), or you regard all that as >> a fundamentally pointless research programme? > When such an approach produces better result that those based on > continuous compact domains, you will let me know? There's a good fellow. Goodness me, *everything* works better for space-time than a compact domain. Compact Lorentz manifolds all exhibit causal pathologies. And we surely need something radiacally different from the usual space-time model built on a diferential manifold if we're going to get a decent theory of quantum gravity. In any case, the various attempts to figure out quantum geometry are trying to find something that reproduces the differential topological/geometrical picture in the appropriate limit. So far, progress is mixed, but with hints that the routes are interesting. Sounds like the 'unaware' possibility that's the right one. > So far Wolfram's work using tessilated automota is a bust. Wolfram's programme bears about as much relation to the stuff I mentioned as chalk does to a cheese sandwich. ==== Subject: Re: Update: Objections to Cantor's Theory > a) The number of naturals is equal to the number of decimal > string representations of the naturals: > Card({ 0, 1, 2, 3, ... }) = > Card({ '0', '1', '2', '3', ... }) > I'll go along with you, but on the condition that you don't throw > away a crucial step in my argument, namely that the above set is > actually _finite_: assume the size of this set is finite and N . What does finite mean when you use it? You have probably covered this before, but I haven't seen it. As I am using finite, N is not finite and cannot be assumed to be finite, no more than we can assume it contains pi. If you want to talk about some set like { 0, 1, 2, 3, ..., N-2, N-1 } that would be fine, but it would not be N. One way to think of that set would be to give the integers a non-standard order, with all the negative integers after all the nonnegative integers, but otherwise the same: { 0, 1, 2, 3, ..., -2, -1 } I'll repeat my argument for reference, without as much verbiage: Card({ 0, 1, 2, 3, ... }) = Card({ '0', '1', '2', '3', ... }) = Card({ '0', '1', '10', '11', ... }) = Card({ '00', '10', '100', '110', ... }) = (note i) Card({ '00', '10', '100', '110', ... }) = (note ii) Card({ '0', '2', '4', '6', ... }) = Card({ 0, 2, 4, 6, ... }) (i) The binary string representations of the naturals with '0' appended on the right (ii) The binary string representations of the even naturals. [...] > g) By steps a, b, c, d, e and f, the number of naturals is > equal to the number of even naturals. > Apart from a minor refinement. See below. Please correct me if I'm wrong, but it looks like you're agreeing with my argument, apart from whether N is finite. But I did not use the infiniteness of N in my argument. I don't think it needs to work for all the finite initial segments, but it does, anyway. What it looks like is that you've let me calculate N one way, then you calculated N another way and got a different answer. They can't all be right, or if they they are, then whatever it is you're calculating is not a property of the set N, but of N plus whichever technique is being used. This strikes me as a very poor model for the idea of size. > [ ... rest deleted ... ] > The number of (all) naturals in the set > { 0, 1, 2, 3, ... , N-1 } > is equal to the number of (even) naturals in the set > { 0, 2, 4, 6, ... , 2(N-1) } > The cardinality of both these sets is N. So far so good. > But Cantorians say more. They say that the second set is > a _subset_ of the first. But this is not true, if you go > along with me in my finitary reasoning. Only part of the > second set, namely > { 0, 2, 4, 6, ... , N-1 } or { 0, 2, 4, 6, ... , N-2 } > is a subset of { 0, 1, 2, 3, ... , N-1 } . > Now take the limit for N -> oo and everything is just fine. > You will not like the conclusion, though. :-( No, I'm fine with the conclusion. This method of comparing initial segments does not give consistent answers. Conclusion: go back to comparing whole sets with bijections. Jim Burns ==== Subject: Re: Update: Objections to Cantor's Theory x is equinumerous with a natural number. > Then 1/2 is not finite. That depends on what '1/2' represents. If it represents a fraction, merely an ordered pair of natural numbers, then it is a finite set. If it is a rational number, as often (usually?) defined as a cetain equivalence class of ordered pairs of integers, then it is an infinite set. Consequences of the usual (?) definitions: A natural number is a finite set. The set of natural numbers is a countably infinite set. An integer is an equivalence class of ordered pairs of natural numbers. An integer is a countably infinite set. The set of integers is a countably infinite set. A rational number is an equivalence class of ordered pairs of integers. A rational number is a countably infinite set. The set of rational numbers is a countably infinite set. A real number is an equivalence class of Cauchy sequences of rational numbers (alternatively, a Dedekind cut of rational numbers). A real number is a countably infinite set. The set of real numbers is an uncountably infinite set. A complex number is an ordered pair of real numbers. A complex number is a finite set. The set of complex numbers is an uncountably infinite set. I hope this stirs no hornets' nests. MoeBlee ==== Subject: Re: Update: Objections to Cantor's Theory Let S be the set of all natural numbers and omega: > S = {0, 1, 2, 3, ..., w} > Do you agree that the union of all members of S is w? > Now let N be the set of all natural numbers. > What is the union of all members of N? > Are you going to say w? > Do you think this a little strange considering that > N doesn't include w and is a proper subset of S? That is a theorem. It's not strange that it's a theorem. I understand why some people do find a theory of infinite sets to be strange. But if one even imagines an abstract world with entities that are themselves infinite, then it would be hard for ANY such world not to have things going on that are not strange to our finite experience and finite apprehension. Morevover, to understand a theory, one does not need to commit to belief in the real existence of such abstract worlds, but rather one may understand that the formal theorems provide rigorous ways of couching the notion of infinity, while it seems that the notion of infinity is needed to express some pretty basic mathematics. So I think challenges about the strangeness of set theory are best directed to realists, not to those who may eschew such ontological commitments. And realists have their arguments too, so there is a meaningful debate. If one can express mathematics without recourse to a notion of infinity, but do so rigorously, then fine with me. But some people would reasonably expect that the treatment be rigorous indeed and, for certain reasons, one would like an alternative theory to have the kind of simplicity in formulation that ZF does have. But if one cannot provide a viable theory without infinite sets, then one's objections about the strangeness of set theory are meaningful only to the extent that they are part of project to actually provide a different foundation. The foundation that set theory provides does have strange things going on with infinity, as I mentioned that it is hard to imagine that infinity is not itself inherently strange from the view of our finite experience. But if this strangeness in the foundation does not interfere with what goes on in the rooms of the house of mathematics, especially the main room of analysis, and, if the foundation does hold up the house, then it's not a bad foundation. And the foundation does hold up the house. At the level of working within a subject such as analysis, don't these mathematical objects as they are constructed all obey the laws we want them to? It may seem strange that the natural number one is defined as the set whose only member is the empty set, but we may do all the mathematics we want to do with the number, no matter the seemingly strange way the number is set theoretically couched. Don't the oridinary notions and theorems about natural numbers and analysis come out just the way they should? That's what would seem to count, especially if viable and radically different alternatives are not available. It's like those clocks and watches that have clear shells. All the crazy looking wiring inside may seem strange, but what matters is that the timekeeping is correct. Some people claim that a foundation is not even needed. But this is a different debate from that of what foundation mathematics should have, given the assumption that a foundation is needed. Note: My remarks are not an endorsement of extreme formalism. Formalism does not demand that mathematics be ONLY symbol manipulation. / > You snipped the part where I define the empty set > to be the smallest ordinal. I didn't realize that that sentence was part of your definition; I thought it was an additional comment about ordinals. I did not mean to misrepresnt your formulation, and regret having inadvertently done so. That said, your definition is still circular. > If you have a better definition, please post it. Definitions are given in the many textbooks on the subject. If you happen to find that my use of the expression 'ordinal' is inconsistent with that of usual set theory, then I'd be happy to be informed. Unless there's a material point at stake, I'm disinclined to play fetch to supply proofs and definitions that are available in any textbook. Your definition is circular, no matter what other definition one would provide. >> Largest ordinal in the set can be defined. >> The largest ordinal is the ordinal that isn't a member of any ordinal in >> the >> set. > Not every set of ordinals has a largest member. > If you have a proof of this, I would love to see it. See below: > The set of natural > numbers is a set of ordinals with no largest member. Every ordinal in > the set of natural numbers is a member of some ordinal in the set of > natural numbers. > Proof? You're asking me to prove that for any natural number there's a greater natural number. In general, you can continue to ask for proofs and definitions until we unpack the theorems all the way back to the axioms and primitives. What's your point? If we're talking about ZF, then please take my remarks about the subject to be in context of first order predicate logic and the ZF axioms and its theorems and usual textbook definitions. If you have something else in mind, then it would help to know what it is. >> The following statement can be proven by induction. Given a set of ordinals, S, the union of S is equal to a member of S. >> (S is a set of ordinals and x is the union of S -> x is a member of S) > If I asked you for such a proof, I might feel self-obliged to correct > it, so I won't ask. Moreover, you gave a circular definition of > 'ordinal', so I have no interest in what you think you've proven based > on what you think ordinals are. > It wasn't circular until you snipped the part about the > empty set being an ordinal. Even with the clause restored, your definition is circular. >> Why isn't the this statement true of all sets of ordinals? > You just said it was. > I did, didn't I. > Since you haven't provided any proofs for your statements, > I guess the statement is correct. Now, THAT is bizarre logic. / Now, back to an earlier matter, you claimed that set theory is contradictory (or, perhaps, 'inconsistent' was your term) and claimed that there is a lack of agreement among set theorists as to what contradiction is. There is hardly a lack of agreement, and I gave you an exact definition. Your claim that set theory is inconsistent is not substantiated by you. MoeBlee ==== Subject: Re: Update: Objections to Cantor's Theory On 26 Jul 2005 13:39:41 -0700, david petry > Our contemporary orthodoxy: To show that there are so-and-sos > is to prove 'So-and-sos exist' from the axioms of set theory. > (Penelope Maddy, Mathematical existence) > Set theory is formal operations on meaningless symobols. So indeed, it > is a fact that we can formally prove the existence of uncountable sets, > where the existential quantifier is a meaningless symbol, and where > uncountable is defined in terms of meaningless symbols. > The whole point I have been trying to make is that all of the mathematics > which has the potential to be useful in understanding our observable world > has meaning, but set theory doesn't. éOn foundations we believe in the reality of mathematics, but of course when philosophers attack us with their paradoxes we rush to hide behind formalism and say Mathematics is just a combination of meaningless symbols, and then we bring out Chapters 1 and 2 on set theory. Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. This sensation is probably an illusion, but is very convenient. That is Bourbaki's attitude toward foundations..82 (Jean Dieudonn.8e) éThe working mathematician is a Platonist on weekdays, a formalist on weekends. On weekdays, when doing mathematics, he's a Platonist, convinced he's dealing with an objective reality whose properties he's trying to determine. On weekends, if challenged to give a philosophical account of the reality, it's easiest to pretend he doesn't believe it. He plays formalist, and pretends mathematics is a meaningless game..82 (R. Hersh) F. ==== Subject: Re: Update: Objections to Cantor's Theory >for N -> oo ? >>Depends on topology. >For heavens sake! What kindof topology can possibly be associated with >something so simple as a sequence of natural numbers? >> Topology is the branch of mathematics where convergence is defined. >> Anytime you are talking about convergence, you are talking about >> topology. You are using that standard topology on the reals to >> calculate the limit of your density. Problem is, I know of no >> standard topology on sets. So, to answer your question about a >> sequence of sets, I need to know what topology you are using. >But aren't the naturals not endowed with a natural topology, I mean >not your shuffled sequence, but just everything in the right order: >1,2,3,4,5,6,7,8,9,10,11,12,13 .. well, you know .. ? >Han de Bruijn Apropos of using topology to define the notion of convergence to oo, we can organize this by considering suitable topologies, first for N in the absence of oo, and then for N U {oo}. 1. Relevant topologies on N: (i) The Alexandroff topology. This treats N as a chain 0<1<2<... It has for its closed sets the initial segments of N (i.e. the open sets are the order filters or up-sets). As such it is T_0 but not T_1; it is also compact even in the absence of oo, trivially because any covering of N by open sets must cover 0 and hence include N itself. (ii) The discrete topology. This doesn't care about order; it has for its open (and closed) sets all subsets of N. This is Hausdorff (T_2) and totally disconnected but not compact. 2. Associated topologies on N U {oo}. (An invariant of the following is that in topologizing this extension of N the subspace topology on N should be left unchanged.) In case (i), there are two possible topologies, Alexandroff and Scott. The Alexandroff topology again has the initial segments for its closed sets. This is equivalent to adjoining oo to every open set of the Alexandroff topology on N, and then (since this turns the empty set into {oo}) putting the empty set back in the topology (so we had to clone the empty set). The Scott topology differs in a single open set: whereas the Alexandroff topology has a finite nonempty open set, namely {oo}, the Scott topology omits this open set. Equivalently, adjoin oo to every nonempty (equivalently, infinite) open set, leaving the empty set untouched (no need to clone any open set). oo is not the limit of 0,1,2,... with the Alexandroff topology, but is with the Scott topology. In case (ii), there are two canonical topologies for N U {oo}, the discrete topology (Hausdorff but not compact) and the Stone-Cech, or one-point, compactification of N by oo (a compact Hausdorff space). The former clones every open set of the discrete topology and adjoins oo to each clone. The latter does no cloning and instead adjoins oo to all the cofinite open sets. Intuitively compactness here means that oo is tucked in tightly to N so that any infinite limiting behavior on N of a continuous function is expressed at oo. 3. Examples Consider functions from NU{oo} to the following spaces. (a) The Sierpinski space S, namely {0,1} with open sets {},{1},{0,1}. Intituitively if f: NU{oo} -> S maps all finite n to 0, then it is continuous if and only if it maps oo to 0. Alexandroff: wrong call (it makes f continuous when f(oo) = 1). Scott: right call: makes f continuous or discontinuous when f(oo) = 0 or 1. Discrete: wrong call (same problem as with Alexandroff) One-point compactification: right call (similar reason to Scott). (b) The reals R with its usual topology. Intuitively if f : NU{oo} -> R maps finite n to 1 - 1/n (a Cauchy sequence), then f is continuous if and only if it maps oo to 1 (instead of to say 1.1 or .9 or 2). Alexandroff and Scott are both wrong here because the inverse image of the open set (.4,.6) is {2} which is not open. So even if f maps oo to 1, both Alexandroff and Scott incorrectly judge f to be discontinuous. (Both are in essence complaining that f is not monotone.) The discrete topology errs in the other direction, judging f to be continuous even when f(oo) = 2. Only the one-point compactification makes the right call on these examples. Hopefully all this makes a little clearer the impact of topology on N U {oo} with regard to limiting behavior. Vaughan Pratt -- Don't contact me at pratt@boole.stanford.edu, substitute cs for boole instead. ==== Subject: Re: Update: Objections to Cantor's Theory > I'm starting a new thread because the other one got > out of control. Well, to add in my interjection: Leibnitz solved the problem of mathematical disagreement, the solution implemented in the 20th century: symbolic logic. So, the bottom line is simple: if there's something wrong with the concept of infinity in set theory then derive A and (not A) via first order predicate calculus from T, where T is a conjunction of all the axioms in a suitable first order formalism of set theory. If not, then there's nothing to discuss and there's nothing wrong with the concept of infinity. There is no room for argument with these or other mathematical issues concerning consistency. There's only room for put up or shut up. The only valid objection to anything is a derivation of A & (not A) in first order predicate calculus. Everything else is just talk and talk is worthless. ==== Subject: Re: Update: Objections to Cantor's Theory > The only valid objection to anything is a derivation of A & (not A) in > first order predicate calculus. Everything else is just talk and talk > is worthless. That's an opinion, not a fact. The mathematics which has the potential to be applied, passes certain reality checks which I have discussed before at length. Cantor's Theory does not pass these reality checks. I see that as a valid objection. ==== Subject: Re: Update: Objections to Cantor's Theory > The only valid objection to anything is a derivation > of A & (not A) in > first order predicate calculus. > Everything else is just talk and talk > is worthless. > That's an opinion, not a fact. THAT is YOUR opinion, and it's factually false. Who the do you think you are, anyway? Where did you get YOUR degree in philosophy of math? The people who actually know the facts about what math is know that this is a fact. You're mathematically ignorant, so you don't know. > The mathematics which has the potential to be applied, A class of math that you cannot define. > passes > certain reality checks Bull. > which I have discussed before at length. Bull. You have never discussed them at all. You have absolutely no conception of what a reality check for a piece of 'math' could possibly ever even be. WE, on the other hand, since WE know what model theory is, DO. > Cantor's Theory does not pass these reality checks. There is NO SUCH THING as Cantor's theory. THat is the first reality check that YOU fail. You cannot define what a check is. For that, you just fail period. In reality. > I see that > as a valid objection. Since you can't even define your checks, and since you don't even know what math is, you cannot have any valid objections. ==== Subject: Re: Update: Objections to Cantor's Theory >>The only valid objection to anything is a derivation of A & (not A) in >>first order predicate calculus. Everything else is just talk and talk >>is worthless. > That's an opinion, not a fact. > The mathematics which has the potential to be applied, passes > certain reality checks which I have discussed before at length. > Cantor's Theory does not pass these reality checks. I see that > as a valid objection. What is a reality check as the term would apply to mathematics. Since mathematical objects have no physical reality (that is because they are abstract objects) it surely cannot be an empirical matter. Do you insist that a mathematical theory must be applicable to the physical world? If so, the geometry of ininitely many dimensional spaces would not pass your muster. How about the theory of non-Archimedian fields. Does that pass a reality check? What about the theory of knots? Only a finite collection of knot types can be realized physically. What about non-Euclidean geometry in its various manifestations? What about the theory of infinite groups? Bob Kolker ==== Subject: Re: Update: Objections to Cantor's Theory > Do you insist that a mathematical theory must be applicable to the > physical world? While I agree that the answer to this question ought to be 'no', you've picked a whole bunch of bits of maths that are in fact applied in physics. > If so, the geometry of ininitely many dimensional spaces > would not pass your muster. Hilbert spaces and Banach manifolds are used in a variety of applications, from QM to field theory and fluids. > How about the theory of non-Archimedian > fields. Depending on just what you mean by that (say, non-standard models of R) there are people who work in dynamical systems and various aspects of mathematical physics including quantum mechanics and kinetic theory who use it. > What about the theory of > knots? See Witten's work on observables in quantum gravity. And in any case, much of knot theory was motivated kind of knots in the ether: this goes back at least to Tait and Thompson. > What about non-Euclidean geometry in its various > manifestations? Navigation on the earth's surface, to be prosaic about it. General relativity, and geometric (classical) mechanics to be a little fancier. > What about the theory of infinite groups? Lie groups are a standard tool in the study of differential equations with symmetries, and gauge theories. But none of this means that the mathematics wasn't interesting or valid before the physicists found out about it. It just shows that it's hard to predict what will turn out to be of physical interest. ==== Subject: Re: Update: Objections to Cantor's Theory > What is a reality check as the term would apply to mathematics. Since > mathematical objects have no physical reality (that is because they are > abstract objects) it surely cannot be an empirical matter. That's _your_ mantra. I have another. And will not repeat it again. > Do you insist that a mathematical theory must be applicable to the > physical world? If so, the geometry of ininitely many dimensional > spaces would not pass your muster. Huh, huh. The whole of Quantum Mechanics is based on ininitely many dimensional spaces. > How about the theory of non-Archimedian > fields. Does that pass a reality check? What about the theory of > knots? Only a finite collection of knot types can be realized > physically. Ask modern physicists, like John Baez (: Google up). > What about non-Euclidean geometry in its various > manifestations? Ask Albert Einstein. (Oh well, he's dead.) > What about the theory of infinite groups? Lie groups? Very useful! I've done those myself. See? Mathematics is even more useful than you thought. Next time, you'd come up with better examples. Han de Bruijn ==== Subject: Regular topological spaces Suppose a topological space has the property that any non-empty open set contains the closure of a non-empty open set. Is the space necessarily regular? Here, T1 is not part of the definition of regularity. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor in Central New Jersey and Manhattan === f30@newsstand.cit.cornell.edu> <85mzoi5wgm.fsf@lola.goethe.zz> <2d6dd$42df72fa$82a1e3ad$12833@news1.tudelft.nl> Discussion, linux) > BTW from the formulation of your answer I assume you're a programmer > and not a mathematician. Vaughan Pratt is a mathematician as well as a computer scientist (and logician). I don't know if one should call him a programmer too. Google is your friend. -- Jesse F. Hughes Love songs suck and losing you ain't worth a damn. -- The poetry of Bad Livers === f30@newsstand.cit.cornell.edu> <85mzoi5wgm.fsf@lola.goethe.zz> Discussion, linux) > There is no need for such a definition. Words are never used > on their own, they are used in *sentences*. The role of a definition > is to understand the meaning of *sentences* involving a word. Well said, Daryl Frege. Seriously, I think this is a crucial point. It would be nice if Tony got it, but what are the chances? -- Jesse F. Hughes Had you told it like it was, it wouldn't be like it is. -- Albert King === f30@newsstand.cit.cornell.edu> <85mzoi5wgm.fsf@lola.goethe.zz> <85fyu9tu92.fsf@lola.goethe.zz> <85hdepqo4q.fsf@lola.goethe.zz> Discussion, linux) > While I'm really enjoying this great thread, at the same time I'm > really bothered that no one has mentioned what to me is by far the > biggest problem with Cantor's theory, namely its claim to > universality. This is certainly the most thoughtful post in this thread. It is a symptom of Usenet, I guess, that it has generated only one other response that I see. The first part of the post (regarding V_4, etc.) seems to me to be the usual structuralist argument that set theory somehow misses the essence by forcing one to pick representations. Something like Benacerraf's point in What numbers cannot be, right? The second part is new to me. It seems to be something like this: 2^A is naturally structured. It is a Boolean algebra. Thus, 2^- should be viewed as a functor from Set to BoolAlg. But then 2^(2^-) only makes sense if the outer 2^- is a functor with domain BoolAlg, and hence should take Boolean algebras B to the set of *Boolean* homomorphisms of B to 2. Here, I guess, you naturally regard 2 as the Boolean algebra (0 < 1). You then deduce that for finite sets A, 2^(2^A) should equal A. For the infinite case, you notice that 2^A is complete and so *there* you interpret 2^(2^A) as the set of *complete* Boolean homomorphisms from 2^A to 2. Again, you get the same result: 2^(2^A) = A, or at least it should[1]. Then you mention the role of forgetful functors and this and that (including Stone duality, which always hurt my poor addled brain). But the niggling point I don't get is the different cases for the finite and infinite sets A. When A is finite, you view 2^A as a functor from Set to BoolAlg and otherwise to CompBoolAlg, but this ad hoc change of codomain seems to strongly undercut the naturalness of your analysis, doesn't it? Without it, you lose the identity, but with it, the whole should seems much weaker: it *should* only if 2^(2^A) retains those features of 2^A which come from A's being finite/infinite[2]. But why those features and not other features of A? Or why *any* features of A per se? Why not just BoolAlg? comments. Footnotes: [1] I don't know the proper notation for should =, but I suppose that it is O(x = y) (where O is the deontic circle). [2] Is there a term like parity which applies to the finite/infinite dichotomy? -- I am a force of Nature. Time is a friend of mine, and We talk about things, here and there. And sometimes We muse a bit [...] and then We watch them go... in the meantime, Time and I, We play with some of them, at least for a little while. --- JSH and His pal, Time. === f30@newsstand.cit.cornell.edu> <85mzoi5wgm.fsf@lola.goethe.zz> <85fyu9tu92.fsf@lola.goethe.zz> <85hdepqo4q.fsf@lola.goethe.zz> <87vf2sj8ma.fsf@phiwumbda.org> Discussion, linux) [...] > But the niggling point I don't get is the different cases for the > finite and infinite sets A. When A is finite, you view 2^A as a > functor from Set to BoolAlg and otherwise to CompBoolAlg, but this ad > hoc change of codomain seems to strongly undercut the naturalness of > your analysis, doesn't it? Without it, you lose the identity, but > with it, the whole should seems much weaker: it *should* only if > 2^(2^A) retains those features of 2^A which come from A's being > finite/infinite[2]. But why those features and not other features of > A? Or why *any* features of A per se? Why not just BoolAlg? I think this is just silliness on my part. In both cases, we could take the codomain to be the category of complete Boolean algebras, right? That would be a uniform treatment of the finite and infinite sets and I suppose it would support that 2^(2^A) = A. -- Jesse F. Hughes I want to really eat myself, so then I'll be a coalgebra. -- Quincy P. Hughes, Age 3 1/2 === > objections to Cantor's Theory, which I plan to contribute > to the Wikipedia. I would be interested in having > *** > While the pure mathematicians almost unanimously accept > Cantor's Theory (with the exception of a small group of > constructivists), there are lots of intelligent people who > believe it to be an absurdity. Typically, these people > are non-experts in pure mathematics, but they are people > who have who have found mathematics to be of great practical > value in science and technology, and who like to view > mathematics itself as a science. What research did you do to determine the size of the GROUPS in question? Did you enumerate them, or are you drawing analogies? > These anti-Cantorians see an underlying reality to > mathematics, namely, computation. They tend to accept the > idea that the computer can be thought of as a microscope > into the world of computation, and mathematics is the > science which studies the phenomena observed through that > microscope. They claim that that paradigm includes all > of the mathematics which has the potential to be applied to > the task of understanding phenomena in the real world (e.g. > in science and engineering). No, we're interested in CHANGING the real world. > Cantor's Theory, if taken seriously, would lead us to believe > that while the collection of all objects in the world of > computation is a countable set, and while the collection of all > identifiable abstractions derived from the world of computation > is a countable set, Bzzzt. There's no limit to the number of stories that can be told. karl m === <42DC3F5F.30904@netscape.net> <85pste8vvp.fsf@lola.goethe.zz> <854qaq8u5i.fsf@lola.goethe.zz> <85mzoi5wgm.fsf@lola.goethe.zz> |If those lengths cannot be infinite, then the set cannot be either. Either you |have an upper bound or you do not, and if there is no upper bound on the values |of the members, then they may be infinite. If not, then what is the upper |bound, and how do you have an infinite set of strings with only finite lengths? Perhaps you're thinking that a restriction on the size has to come in the form of a dividing line, with the allowed lengths being on one side. There's a restriction, but not based on a dividing line. Finite lengths are allowed, but infinite lengths are not allowed. Since there's nothing in between finite and infinite, there's no point at which to draw a dividing line. There is, nevertheless, a conceptual distinction between finite and infinite, which makes it possible to allow all finite lengths but forbid all infinite lengths. Keith Ramsay === >Daryl McCullough said: >> Tony Orlow says... >despite the fact that an infinite set of whole numbers requires >infinite whole numbers >> That's false, no matter how many times you say it. No finite >> set can contain every (finite) natural. Why? Because every finite >> set of naturals has a largest element, and there is no largest >> finite natural. >That's false, no matter how many times you say it. Which part do you disagree with? Do you disagree that every finite set of naturals has a largest element or do you disagree that there is no largest finite natural? Alan -- Defendit numerus === said: >>Daryl McCullough said: > Tony Orlow says... >despite the fact that an infinite set of whole numbers requires >>infinite whole numbers That's false, no matter how many times you say it. No finite > set can contain every (finite) natural. Why? Because every finite > set of naturals has a largest element, and there is no largest > finite natural. >>That's false, no matter how many times you say it. > Which part do you disagree with? Do you disagree that every finite > set of naturals has a largest element or do you disagree that there > is no largest finite natural? Don't expect an answer. Daryl has pointed out this overt contradiction in TO's views in several posts, all ignored. It gets much too clearly and succinctly to the heart of the matter. === Barb Knox said: >> , >keep in mind thatinductive proof IS an infinite loop, >so that incrementing in the loop createsinfinite values, >and the quality of finiteness is not maintained >over thoseinfinite iterations of the loop. >> Using your computational view, consider the following >> infinite loop (using some unbounded-precision arithmetic >> system similar to java.math.BigInteger): >> for (i = 0; ; i++) { >> println(i); >> } >> Now, although this is an INFINITE loop, every value printed will >> be FINITE. Right? >In any case, sure, the program will spit out finite numbers, > Right. >snce it is a finite machine running in finite time. > Wrong. Just like your notion of infinite loop over the naturals is > not encumbered by limited memory, neither is this program. Consider it > to be running on a machine with unbounded memory, so it could literally > run forever. >If the machine had infinite capacity > Yes, that's the appropriate model. >and infinite funtime, it could conceivably produce infinite results. > HOW?? No matter how long it runs, EVERY printed value is finite. > WHEN EXACTLY do you think it would start producing infinite values > (whatever those might look like). At the point that the runtime became infinite, which is obviously not an identifiable point. At what point DOES the runtime become infinite? A million years? a billion? If you have infinite runtime, starting with a finite amount of time, then you have infinite numbers, starting with finite ones. You can't have it both ways. -- Smiles, Tony === > Yes, that's the appropriate model. and infinite funtime, it could conceivably produce infinite results. HOW?? No matter how long it runs, EVERY printed value is finite. > WHEN EXACTLY do you think it would start producing infinite values > (whatever those might look like). > At the point that the runtime became infinite, which is obviously not an > identifiable point. Never! But the prinout will, in finite time exceed any preassigned finite value, which is precisely what is meant by unbounded. === >Barb Knox said: >> , >keep in mind thatinductive proof IS an infinite loop, >so that incrementing in the loop createsinfinite values, >and the quality of finiteness is not maintained >over thoseinfinite iterations of the loop. >> Using your computational view, consider the following >> infinite loop (using some unbounded-precision arithmetic >> system similar to java.math.BigInteger): >> for (i = 0; ; i++) { >> println(i); >> } >> Now, although this is an INFINITE loop, every value >> printed will be FINITE. Right? >In any case, sure, the program will spit out finite numbers, >> Right. >snce it is a finite machine running in finite time. >> Wrong. Just like your notion of infinite loop over the naturals is >> not encumbered by limited memory, neither is this program. Consider it >> to be running on a machine with unbounded memory, so it could literally >> run forever. >If the machine had infinite capacity >> Yes, that's the appropriate model. >and infinite funtime, it could conceivably produce infinite results. >> HOW?? No matter how long it runs, EVERY printed value is finite. >> WHEN EXACTLY do you think it would start producing infinite values >> (whatever those might look like). >At the point that the runtime became infinite, which is obviously not an >identifiable point. At what point DOES the runtime become infinite? A million >years? a billion? If you have infinite runtime, starting with a finite amount >of time, then you have infinite numbers, starting with finite ones. You can't >have it both ways. It sounds like you're saying that it runs for an infinite amount of time (producing all the finite naturals) AND THEN it starts producing infinite ones. Right? -- --------------------------- | BBB b Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- === <85hdepqo4q.fsf@lola.goethe.zz> > , >keep in mind thatinductive proof IS an infinite loop, >so that incrementing in the loop createsinfinite values, >and the quality of finiteness is not maintained >over thoseinfinite iterations of the loop. >> Using your computational view, consider the following >> infinite loop (using some unbounded-precision arithmetic >> system similar to java.math.BigInteger): >> for (i = 0; ; i++) { >> println(i); >> } >> Now, although this is an INFINITE loop, every value printed will >> be FINITE. Right? >In any case, sure, the program will spit out finite numbers, > Right. >snce it is a finite machine running in finite time. > Wrong. Just like your notion of infinite loop over the naturals is > not encumbered by limited memory, neither is this program. Consider it > to be running on a machine with unbounded memory, so it could literally > run forever. >If the machine had infinite capacity > Yes, that's the appropriate model. >and infinite funtime, it could conceivably produce infinite results. > HOW?? No matter how long it runs, EVERY printed value is finite. > WHEN EXACTLY do you think it would start producing infinite values > (whatever those might look like). > At the point that the runtime became infinite, which is obviously not an > identifiable point. At what point DOES the runtime become infinite? A million > years? a billion? If you have infinite runtime, starting with a finite amount > of time, then you have infinite numbers, starting with finite ones. You can't > have it both ways. There is no need for infinite run times. You are assuming that each step requires a CONSTANT amount of time. Mathematics makes no such requirement. See Zeno's paradox. Each of the values that the java program produces is finite, yet there are an infinite number of them. karl m === !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > At the point that the runtime became infinite, which is obviously > not an identifiable point. At what point DOES the runtime become > infinite? A million years? a billion? If you have infinite runtime, > starting with a finite amount of time, then you have infinite > numbers, starting with finite ones. You can't have it both ways. I prefer to use induction and get my proofs finished before the end of time. -- === Virgil said: > Virgil said: Virgil said: > , Virgil occasionally needs his ass kicked. Maybe I do too, > from time to time. Such is life. TO seems to take such offense at my proofs of his manifold > errors and delusions as to want to kick my ass. If TO could do it by showing my logic flawed, he would have > done it long since. > Been there, done that, without response from you. > > I have no idea where TO thinks he has been, but he has not been > where he > claims, except in the delusions of his own mind. > You mean you entirely disregarded any flaws I pointed out, as you > admit, below. > To points out what he thinks are flaws, but which are only part of > hi8s own delusions. You tried to prove that the > paths in an infinite tree were uncountable but the branches were > countable > I did prove it. That TO does not like my proof does not invalidate it. > > Since the only objections to my proof were the equally false > counteres and misrepresentations by TO and WM, and my proofs were > vetted by and approved by others less mathematically incompetent > than either TO and WM, I regard those proofs as being validated by > those who know how. > I don't think anyone else commented on them, and you never repsonded > when I pointed out that you were switching trees and being dishonest > in the proof. > Since I was at all times dealing with a maximal binary tree, and as all > such are tree isomorphic, there is no way that I could have switched > trees. You switched interpretations of the tree, using a tree where the paths were digital numbers to prove the nodes were countable, and using a tree where the paths were subsets of the naturals to prove the paths were uncountable, based on the notion that a power set of the naturals is not countable, which is nonsense anyway. > If TO prefers it, I could rephrase the proof so that each branch is > a finite string of binary bits and each path is an infinite string > of binary bits. The same conclusion holds in either form, that the > set of branches (or nodes) is countably infinite set but the set > of paths is not. > Why don't you try it with a SINGLE tree, rather than using two > different trees? Try it with a branch being a bit and a path being a > bitstring, or a branch denoting membership of an element in a path > representing a subset. Just don't chnage trees in the middle of your > proof, and you may get a correct result. There are precisely half as > many paths as branches in any complete binary tree, finite or > infinite. It's this kind of nonsensical conclusion that demonstrates > the inconsistencies of set theory. > Theorem: In a maximal binary tree, the set of nodes and the set of > branches are both counably infinite but the set of paths is uncountably > infinite. An INFINITE maximal binary tree, I assume you mean. > Proof: > In a maximal binary tree, every node except the root is the terminus of > exactly one branch, and the position of every node except the root node > can be indicated by a finite sequence of L's for left branches and R's > for right branches connecting the root node to that given node. This > same string identifies the branch terminating in that given node. So you have the same number of branches as nodes, minus 1. You say each path is finite, but that is not the case in an infinite binary tree. Every path has an infinite number of nodes and branches. > This gives a 1 to 1 correspondence between the finite strings of L's and > R's and the branches (or the non-root nodes). Such sets of finite > strings can be countable. Did you just draw a bijection between the branches and the paths? That must have been a mistake, because it is correct (except there are twice as many branches as paths). > We do the same thing, with the same tree, for paths, except that since > no path has an ending node or a last branch, each path-string is of > infinite length. Um, wait a minute. Did you not say every node except the root node can be indicated by a FINITE sequence of L's for left branches and R's for right branches connecting the root node to that given node? Aren't those the paths in the tree, which you said were finite sequences of branches? And now, suddenly, they're infinite? Nice trick. > The set of finite branch-strings is countably infinite. > The set of infinite path-strings is infinite but not countable. So strings of branches are different from paths? Interesting. Care to explain the difference? > Q.E.D. Yeah if quid erat demonstrandum erat stupiditas tui. (yeah I took Latin, after Spanish, and before Greek) >It's a dishonest > mathematical trick, and typical in this area, whether conscious or > not. > It is mathematical, and might be called a trick, since it is so easy, > but it is straightforward, and not in the least dishonest, and it does > precisely what I said it would do, show that the set of branches (or, > equivalently, the set of nodes) is countably infinite but the set of > paths is infinite but uncountable. Sure, if the paths are finite, and then suddenly infinite. That sure makes it easy. Oh, and not at all dishonest. Right. > That TO does not like it to be so obviously wrong, should give him pause > in future. If you could offer just ONE honest proof or statement, I wouldn't consider you a troll. -- Smiles, Tony === > Virgil said: Virgil said: > Since I was at all times dealing with a maximal binary tree, and as all > such are tree isomorphic, there is no way that I could have switched > trees. > You switched interpretations of the tree, using a tree where the paths were > digital numbers to prove the nodes were countable, and using a tree where the > paths were subsets of the naturals to prove the paths were uncountable, based > on the notion that a power set of the naturals is not countable, which is > nonsense anyway. If the base was the same tree, that is all that matters. The fact that I used different proofs to prove different things hardly constitutes a valid objection. What TO really is objecting to is the validity of both proofs. I really did prove that the set of branches (or the set of nodes) of a maximal binary tree are denumerable but that the set ofpaths is not. THAT is what TO objects to. If TO prefers it, I could rephrase the proof so that each branch is > a finite string of binary bits and each path is an infinite string > of binary bits. The same conclusion holds in either form, that the > set of branches (or nodes) is countably infinite set but the set > of paths is not. Why don't you try it with a SINGLE tree, rather than using two > different trees? Try it with a branch being a bit and a path being a > bitstring, or a branch denoting membership of an element in a path > representing a subset. Just don't chnage trees in the middle of your > proof, and you may get a correct result. There are precisely half as > many paths as branches in any complete binary tree, finite or > infinite. It's this kind of nonsensical conclusion that demonstrates > the inconsistencies of set theory. Theorem: In a maximal binary tree, the set of nodes and the set of > branches are both counably infinite but the set of paths is uncountably > infinite. > An INFINITE maximal binary tree, I assume you mean. > > Proof: > In a maximal binary tree, every node except the root is the terminus of > exactly one branch, and the position of every node except the root node > can be indicated by a finite sequence of L's for left branches and R's > for right branches connecting the root node to that given node. This > same string identifies the branch terminating in that given node. > So you have the same number of branches as nodes, minus 1. You say > each path is finite, but that is not the case in an infinite binary > tree. What I said, for those who can read better than TO, is that the sequence of branches and nodes between the root node and any other node is finite. Since in a maximal binary tree no complete path ends in a node (or ends at all) This finiteness does not apply to any such path. This gives a 1 to 1 correspondence between the finite strings of L's and > R's and the branches (or the non-root nodes). Such sets of finite > strings can be countable. > Did you just draw a bijection between the branches and the paths? No! Try reading what I actually say, TO, rather than what you want me to have said. We do the same thing, with the same tree, for paths, except that since > no path has an ending node or a last branch, each path-string is of > infinite length. > Um, wait a minute. Why? Just because TO can't read is no reason for anybody else to stop. Did you not say every node except the root node can be > indicated by a FINITE sequence of L's for left branches and R's for right > branches connecting the root node to that given node? Aren't those the paths > in the tree NO! A path in a maximal binary tree does not have any terminal or leaf node, it goes on forever, just like the tree of which is is a part. The set of finite branch-strings is countably infinite. > The set of infinite path-strings is infinite but not countable. > So strings of branches are different from paths? Interesting. Care to explain > the difference? A finite string of branches has an ending node, but a path does not. Q.E.D. >It's a dishonest > mathematical trick, and typical in this area, whether conscious or > not. It is mathematical, and might be called a trick, since it is so easy, > but it is straightforward, and not in the least dishonest, and it does > precisely what I said it would do, show that the set of branches (or, > equivalently, the set of nodes) is countably infinite but the set of > paths is infinite but uncountable. > Sure, if the paths are finite, and then suddenly infinite. I have been very careful not to speak of paths when I meant finite chains of branches and nodes. That TO still sees paths where none exist may be due to his excessive self-medication. === <42DD6183.5030705@netscape.net> self-medication. Medication is administered by doctors' order, where the final arbitration of excessive or adequate lay (with the doctor, not the patient). The only thing self induced is FOOD, as in FOOD-FOR-THOUGHT. karl m === Robert Low said: > No, I misread those statements. The first, if you are referring to my > arguments, should be that there is an infinite set of whole numbers, and the > second is that there is an infinite whole number in the set. Those two > statements imply each other because of the constant finite difference between > whole numbers. > OK, so how many elements are there in the set of all finite > natural numbers? Some finite, indeterminate number. You tell me the largest finite number, and that's the set size. It doesn't exist? Well, then, I can't help you. -- Smiles, Tony === > Robert Low said: No, I misread those statements. The first, if you are referring to my > arguments, should be that there is an infinite set of whole numbers, and > the > second is that there is an infinite whole number in the set. Those two > statements imply each other because of the constant finite difference > between > whole numbers. OK, so how many elements are there in the set of all finite > natural numbers? Some finite, indeterminate number. You tell me the largest finite number, and > that's the set size. It doesn't exist? Well, then, I can't help you. If it doesn't exist, there is no finite set size to the set of all finitely generated naturals. Since TO is the only one declaring that there is any finite bound on the number of finitely generated naturals, it is TO's, and only TO's, problem to produce such a bound, not that of those who reject that claim. Does TO's argue that because we do not provide him what we claim does not exist, it must exist? About par for TO's level of competence. === > Robert Low said: >>OK, so how many elements are there in the set of all finite >>natural numbers? > Some finite, indeterminate number. You tell me the largest finite number, and > that's the set size. It doesn't exist? Well, then, I can't help you. In fact, you can't even make sense. === >Robert Low said: >> OK, so how many elements are there in the set of all finite >> natural numbers? >Some finite, indeterminate number. >You tell me the largest finite number, and >that's the set size. So you really think that there is some number n such that n is finite, but if you add 1 you get an infinite number? Maybe it's 7? Maybe 7 is the largest finite number, and 8 is actually infinite? >It doesn't exist? Well, then, I can't help you. In fact, a set is finite if and only if the number of elements is equal to a natural number. There is no largest natural number, and there is no largest finite set. The collection of all finite natural numbers is an infinite set. -- Daryl McCullough Ithaca, NY === Daryl McCullough said: >Robert Low said: >> OK, so how many elements are there in the set of all finite >> natural numbers? >Some finite, indeterminate number. >You tell me the largest finite number, and >that's the set size. > So you really think that there is some number n such that n is > finite, but if you add 1 you get an infinite number? (sigh) This is the last time I answer this question NOOOOOOOOOOO!!!!!!!!! > Maybe it's 7? Maybe 7 is the largest finite number, and > 8 is actually infinite? Don't be stupid. >It doesn't exist? Well, then, I can't help you. See? I already answered the question. > In fact, a set is finite if and only > if the number of elements is equal to a natural number. > There is no largest natural number, and there is no > largest finite set. The collection of all finite natural > numbers is an infinite set. The set of all finite numbers up to a given number has that number in it, which is also the set size. Any subset of N has a size that is in N. > -- -- Smiles, Tony === > Daryl McCullough said: Robert Low said: > OK, so how many elements are there in the set of all finite >> natural numbers? >Some finite, indeterminate number. You tell me the largest finite >number, and that's the set size. So you really think that there is some number n such that n is > finite, but if you add 1 you get an infinite number? > (sigh) This is the last time I answer this question > NOOOOOOOOOOO!!!!!!!!! But since every natural has an immediate successor, and except for the first. an immediate predecessor, and there are no gaps( at least if one accepts Peano), the only way of getting from finite to infinite is by adding 1 to some finite natural to get an infinite natural. > Maybe it's 7? Maybe 7 is the largest finite number, and 8 is > actually infinite? > Don't be stupid. He is just trying to come down to your level, TO. In fact, a set is finite if and only if the number of elements is > equal to a natural number. There is no largest natural number, and > there is no largest finite set. The collection of all finite > natural numbers is an infinite set. > The set of all finite numbers up to a given number has that number in > it, which is also the set size. Any subset of N has a size that is in > N. What member of N is the size of N? How about the size of N{1}? The size of the set of even naturals or the size of the set of odd naturals? For the standard theory these all have trivially easy answers, and none of the sizes are members of N. For TO's theory it depends on how his medicatins are affecting him that day. === >> Daryl McCullough said: Robert Low said: > OK, so how many elements are there in the set of all finite >> natural numbers? >Some finite, indeterminate number. You tell me the largest finite >number, and that's the set size. So you really think that there is some number n such that n is > finite, but if you add 1 you get an infinite number? >> (sigh) This is the last time I answer this question >> NOOOOOOOOOOO!!!!!!!!! > But since every natural has an immediate successor, and except for the > first. an immediate predecessor, and there are no gaps( at least if one > accepts Peano), the only way of getting from finite to infinite is by > adding 1 to some finite natural to get an infinite natural. > Maybe it's 7? Maybe 7 is the largest finite number, and 8 is > actually infinite? >> Don't be stupid. > He is just trying to come down to your level, TO. In fact, a set is finite if and only if the number of elements is > equal to a natural number. There is no largest natural number, and > there is no largest finite set. The collection of all finite > natural numbers is an infinite set. >> The set of all finite numbers up to a given number has that number in >> it, which is also the set size. Any subset of N has a size that is in >> N. > What member of N is the size of N? How about the size of N{1}? The size > of the set of even naturals or the size of the set of odd naturals? > For the standard theory these all have trivially easy answers, and none > of the sizes are members of N. For TO's theory it depends on how his > medicatins are affecting him that day. What member of N is the color of N? How about the color of N{1}? The color of the set of even naturals or the color of the set of odd naturals? === >> Daryl McCullough said: Robert Low said: > OK, so how many elements are there in the set of all finite >> natural numbers? >Some finite, indeterminate number. You tell me the largest finite >number, and that's the set size. So you really think that there is some number n such that n is > finite, but if you add 1 you get an infinite number? >> (sigh) This is the last time I answer this question >> NOOOOOOOOOOO!!!!!!!!! > But since every natural has an immediate successor, and except for the > first. an immediate predecessor, and there are no gaps( at least if one > accepts Peano), the only way of getting from finite to infinite is by > adding 1 to some finite natural to get an infinite natural. > Maybe it's 7? Maybe 7 is the largest finite number, and 8 is > actually infinite? >> Don't be stupid. > He is just trying to come down to your level, TO. In fact, a set is finite if and only if the number of elements is > equal to a natural number. There is no largest natural number, and > there is no largest finite set. The collection of all finite > natural numbers is an infinite set. >> The set of all finite numbers up to a given number has that number in >> it, which is also the set size. Any subset of N has a size that is in >> N. > What member of N is the size of N? How about the size of N{1}? The size > of the set of even naturals or the size of the set of odd naturals? > For the standard theory these all have trivially easy answers, and none > of the sizes are members of N. For TO's theory it depends on how his > medicatins are affecting him that day. > What member of N is the color of N? How about the color of N{1}? The > color of the set of even naturals or the color of the set of odd naturals? TO says above, and I quote, Any subset of N has a size that is in N. So TO raises the issue of whether certain sizes are in N or not. As far as I can see, PJ is the first and only person to mention color in connection with properties of sets. So perhaps PJ should be the one to answer his own questions, seeing that he seems to be the only one whom they interest. === > TO says above, and I quote, Any subset of N has a size that is in N. > So TO raises the issue of whether certain sizes are in N or not. Have you received the polemic on this yet? > As far as I can see, PJ is the first and only person to mention color in > connection with properties of sets. So perhaps PJ should be the one to > answer his own questions, seeing that he seems to be the only one whom > they interest. Is Mr Joker satisfied with the outcome of his NOMINATION? You got a hearty I CONCUR from Mr Virgil. karl m === >> OK, so how many elements are there in the set of all finite >> natural numbers? >Some finite, indeterminate number. You tell me the largest finite >number, and that's the set size. So you really think that there is some number n such that n is > finite, but if you add 1 you get an infinite number? >> (sigh) This is the last time I answer this question >> NOOOOOOOOOOO!!!!!!!!! But since every natural has an immediate successor, and except for the > first. an immediate predecessor, and there are no gaps( at least if one > accepts Peano), the only way of getting from finite to infinite is by > adding 1 to some finite natural to get an infinite natural. Maybe it's 7? Maybe 7 is the largest finite number, and 8 is > actually infinite? >> Don't be stupid. > He is just trying to come down to your level, TO. In fact, a set is finite if and only if the number of elements is > equal to a natural number. There is no largest natural number, and > there is no largest finite set. The collection of all finite > natural numbers is an infinite set. > The set of all finite numbers up to a given number has that number in >> it, which is also the set size. Any subset of N has a size that is in >> N. What member of N is the size of N? How about the size of N{1}? The size > of the set of even naturals or the size of the set of odd naturals? > For the standard theory these all have trivially easy answers, and none > of the sizes are members of N. For TO's theory it depends on how his > medicatins are affecting him that day. > What member of N is the color of N? How about the color of N{1}? The > color of the set of even naturals or the color of the set of odd naturals? > TO says above, and I quote, Any subset of N has a size that is in N. > So TO raises the issue of whether certain sizes are in N or not. > As far as I can see, PJ is the first and only person to mention color in > connection with properties of sets. So perhaps PJ should be the one to > answer his own questions, seeing that he seems to be the only one whom > they interest. Actually, Virgil, coloring trees and their parts is a useful concept in graph theory. karl m === > Actually, Virgil, coloring trees and their parts is a useful concept in > graph theory. karl m coloring trees is simply applying labels. It has nothing to do inherently with frequency of electromagnetic radiation. The colors used in graph theory are simply conventional labels. Bob Kolker === <85ll43dvil.fsf@lola.goethe.zz> f65@newsstand.cit.cornell.edu> <3knnm8FuvsriU2@individual.net> OK, so how many elements are there in the set of all finite >> natural numbers? >Some finite, indeterminate number. You tell me the largest finite >number, and that's the set size. So you really think that there is some number n such that n is > finite, but if you add 1 you get an infinite number? > (sigh) This is the last time I answer this question > NOOOOOOOOOOO!!!!!!!!! > But since every natural has an immediate successor, and except for the > first. an immediate predecessor, and there are no gaps( at least if one > accepts Peano), the only way of getting from finite to infinite is by > adding 1 to some finite natural to get an infinite natural. > Maybe it's 7? Maybe 7 is the largest finite number, and 8 is > actually infinite? > Don't be stupid. > He is just trying to come down to your level, TO. Don't DO THAT, and REVERSE IT EACH AND EVERY TIME IT OCCURS. Tony's contradiction was ALREADY CAUGHT when he said that the number of natural numbers is an infinite-indeterminate number. karl m === >Robert Low said: > OK, so how many elements are there in the set of all finite >> natural numbers? Tony replied. >Some finite, indeterminate number. That is an out-and-out contradiction. Let FN be the collection of all finite natural numbers. You say that FN is finite. You say that that means that its size is equal to some finite natural number. So call that number L. If L is finite, then it must be an element of FN, because FN is the collection of *all* finite natural numbers. But that means that FN contains at least L+1 elements: 0, 1, 2, ..., L. That contradicts the claim that FN contains exactly L elements. Your theory is self-contradictory. Not that *you* would ever notice the contradiction, because you are just making things up as you go. You are just playing, not caring whether what you're saying makes sense or not. -- Daryl McCullough Ithaca, NY === >>Robert Low said: > OK, so how many elements are there in the set of all finite > natural numbers? > Tony replied. >>Some finite, indeterminate number. > That is an out-and-out contradiction. Let FN be the > collection of all finite natural numbers. You say that > FN is finite. You say that that means that its size is > equal to some finite natural number. He never said that. He said Some finite, indeterminate number. He didn't say Some finite, indeterminate natural number. But even if he did, mathematicians have their own meaning of words and therefore his natural numbers might be different than mathematicians. > So call that number > L. If L is finite, then it must be an element of FN, > because FN is the collection of *all* finite natural > numbers. But that means that FN contains at least L+1 > elements: 0, 1, 2, ..., L. That contradicts the claim > that FN contains exactly L elements. Too bad he didn't imply that stuff. > Your theory is self-contradictory. Not that *you* would > ever notice the contradiction, because you are just making > things up as you go. You are just playing, not caring whether > what you're saying makes sense or not. I think you are the one that is trying to put nonsense into his post. > -- === > OK, so how many elements are there in the set of all finite > natural numbers? > Tony replied. >>Some finite, indeterminate number. > That is an out-and-out contradiction. Let FN be the > collection of all finite natural numbers. You say that > FN is finite. You say that that means that its size is > equal to some finite natural number. > He never said that. He said Some finite, indeterminate > number. He didn't say Some finite, > indeterminate natural number. But even if he did, > mathematicians have their own meaning of words > and therefore his natural numbers might be different than > mathematicians. Words have an agreed meaning. That's why we have dictionaries. Mathematicians learn to use words like everyone else. karl m === >>Robert Low said: > OK, so how many elements are there in the set of all finite > natural numbers? > Tony replied. >>Some finite, indeterminate number. > That is an out-and-out contradiction. Let FN be the > collection of all finite natural numbers. You say that > FN is finite. You say that that means that its size is > equal to some finite natural number. > He never said that. He said Some finite, indeterminate > number. He didn't say Some finite, > indeterminate natural number. But even if he did, > mathematicians have their own meaning of words > and therefore his natural numbers might be different than > mathematicians. But TO insists that OUR natural numbers have to follow HIS rules, despite the fact that his rules contradict our rules on many issues. > So call that number > L. If L is finite, then it must be an element of FN, > because FN is the collection of *all* finite natural > numbers. But that means that FN contains at least L+1 > elements: 0, 1, 2, ..., L. That contradicts the claim > that FN contains exactly L elements. > Too bad he didn't imply that stuff. > Your theory is self-contradictory. Not that *you* would > ever notice the contradiction, because you are just making > things up as you go. You are just playing, not caring whether > what you're saying makes sense or not. > I think you are the one that is trying to put nonsense into his > post. No, just the one pointing it out! It was already there. There is a good deal of nonsense n all of TO's attempts to reformulate mathematics in his own image. === >>Robert Low said: > OK, so how many elements are there in the set of all finite > natural numbers? Tony replied. >Some finite, indeterminate number. That is an out-and-out contradiction. Let FN be the > collection of all finite natural numbers. You say that > FN is finite. You say that that means that its size is > equal to some finite natural number. >> He never said that. He said Some finite, indeterminate >> number. He didn't say Some finite, >> indeterminate natural number. But even if he did, >> mathematicians have their own meaning of words >> and therefore his natural numbers might be different than >> mathematicians. > But TO insists that OUR natural numbers have to follow HIS rules, > despite the fact that his rules contradict our rules on many issues. Then why argue with him. Tell him you use different rules and uncommon meanings of words and be done with it. > So call that number > L. If L is finite, then it must be an element of FN, > because FN is the collection of *all* finite natural > numbers. But that means that FN contains at least L+1 > elements: 0, 1, 2, ..., L. That contradicts the claim > that FN contains exactly L elements. >> Too bad he didn't imply that stuff. > Your theory is self-contradictory. Not that *you* would > ever notice the contradiction, because you are just making > things up as you go. You are just playing, not caring whether > what you're saying makes sense or not. >> I think you are the one that is trying to put nonsense into his >> post. > No, just the one pointing it out! It was already there. There is a good > deal of nonsense n all of TO's attempts to reformulate mathematics in > his own image. He's using common word meanings. You are using (admittedly by mathematicians) uncommon word meanings. === !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > Daryl McCullough said: >Robert Low said: > OK, so how many elements are there in the set of all finite >> natural numbers? >Some finite, indeterminate number. >You tell me the largest finite number, and >that's the set size. >> So you really think that there is some number n such that n is >> finite, but if you add 1 you get an infinite number? > (sigh) This is the last time I answer this question > NOOOOOOOOOOO!!!!!!!!! >> Maybe it's 7? Maybe 7 is the largest finite number, and >> 8 is actually infinite? > Don't be stupid. >It doesn't exist? Well, then, I can't help you. > See? I already answered the question. >> In fact, a set is finite if and only >> if the number of elements is equal to a natural number. >> There is no largest natural number, and there is no >> largest finite set. The collection of all finite natural >> numbers is an infinite set. > The set of all finite numbers up to a given number has that number > in it, which is also the set size. Correct. > Any subset of N has a size that is in N. Incorrect. The size of the set of positive even numbers, for example, is not in N. -- === >>OK, so how many elements are there in the set of all finite >>natural numbers? >Some finite, indeterminate number. >You tell me the largest finite number, and >that's the set size. >>So you really think that there is some number n such that n is >>finite, but if you add 1 you get an infinite number? > (sigh) This is the last time I answer this question > NOOOOOOOOOOO!!!!!!!!! Or then again... So your claim that FN is finite implies that there is some number n that is the largest finite natural number. That means that n is finite n+1 is infinite By any sane definition of infinite I would think that would be impossible. And you replied: ... In the case of the finite naturals, it simply does not hold true. So is that NOOOOOOOOO! or YESSSSSSSSSSS! ? (I know it's more convenient if it's both, depending on what you're claiming at any given moment, but that's cheating.) > The set of all finite numbers up to a given number has that number in it, which > is also the set size. Any subset of N has a size that is in N. So which element of N is the size of the set of even numbers? Hang on a minute...there something sticking into my gum...I'll just pull it out. Well, I'll be damned, it's a hook! How did that get there? === 2d@newsstand.cit.cornell.edu> f65@newsstand.cit.cornell.edu> <3knnm8FuvsriU2@individual.net> No, I misread those statements. The first, if you are referring to my > arguments, should be that there is an infinite set of whole numbers, and the > second is that there is an infinite whole number in the set. Those two > statements imply each other because of the constant finite difference between > whole numbers. > OK, so how many elements are there in the set of all finite > natural numbers? > Some finite, indeterminate number. You tell me the largest finite number, and > that's the set size. It doesn't exist? Well, then, I can't help you. De*termi*nate (?), a. [L. determinatus, p. p. of determinare. See Determine.] 1. Having defined limits; not uncertain or arbitrary; fixed; established; definite. Thus indeterminate is the exact opposite of fine. You can't have it both ways. karl m === malbrain@yahoo.com said: > Robert Low said: No, I misread those statements. The first, if you are referring to my > arguments, should be that there is an infinite set of whole numbers, and the > second is that there is an infinite whole number in the set. Those two > statements imply each other because of the constant finite difference between > whole numbers. OK, so how many elements are there in the set of all finite > natural numbers? Some finite, indeterminate number. You tell me the largest finite number, and > that's the set size. It doesn't exist? Well, then, I can't help you. > De*termi*nate (?), a. [L. determinatus, p. p. of determinare. See > Determine.] > 1. Having defined limits; not uncertain or arbitrary; fixed; > established; definite. > Thus indeterminate is the exact opposite of fine. You can't have it > both ways. karl m Indeetrminate means the opposite, so: without defined limits; uncertain or arbitrary; not fixed; unestablished; indefinite. That sounds about right, especially the first definition. -- Smiles, Tony === No, I misread those statements. The first, if you are referring to my > arguments, should be that there is an infinite set of whole numbers, and the > second is that there is an infinite whole number in the set. Those two > statements imply each other because of the constant finite difference between > whole numbers. OK, so how many elements are there in the set of all finite > natural numbers? Some finite, indeterminate number. You tell me the largest finite number, and > that's the set size. It doesn't exist? Well, then, I can't help you. > De*termi*nate (?), a. [L. determinatus, p. p. of determinare. See > Determine.] > 1. Having defined limits; not uncertain or arbitrary; fixed; > established; definite. > Thus indeterminate is the exact opposite of fine. You can't have it > both ways. karl m > Indeetrminate means the opposite, so: without defined limits; uncertain or > arbitrary; not fixed; unestablished; indefinite. > That sounds about right, especially the first definition. Then you agree that the number of natural numbers is not both finite and indeterminate since they mean the opposite, and that this number is up-for-grabs? (in 1913)? karl m === 2d@newsstand.cit.cornell.edu> f65@newsstand.cit.cornell.edu> <3knnm8FuvsriU2@individual.net> that's the set size. It doesn't exist? Well, then, I can't help you. > De*termi*nate (?), a. [L. determinatus, p. p. of determinare. See > Determine.] > 1. Having defined limits; not uncertain or arbitrary; fixed; > established; definite. > Thus indeterminate is the exact opposite of fine. You can't have it > both ways. Ooops. indeterminate is the exact opposite of finite. You've uncovered a contradiction about the count of elements in an infinite set that cannot be resolved from the definitions of finite and indeterminate. karl m === !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > Some finite, indeterminate number. You tell me the largest finite number, and > that's the set size. It doesn't exist? Well, then, I can't help you. >> De*termi*nate (?), a. [L. determinatus, p. p. of determinare. See >> Determine.] >> 1. Having defined limits; not uncertain or arbitrary; fixed; >> established; definite. >> Thus indeterminate is the exact opposite of fine. You can't have it >> both ways. > Ooops. indeterminate is the exact opposite of finite. You've > uncovered a contradiction about the count of elements in an infinite > set that cannot be resolved from the definitions of finite and > indeterminate. karl m Uh, no. Indeterminate just means unspecified, not infinite. -- === > Some finite, indeterminate number. You tell me the largest finite number, and > that's the set size. It doesn't exist? Well, then, I can't help you. >> De*termi*nate (?), a. [L. determinatus, p. p. of determinare. See >> Determine.] 1. Having defined limits; not uncertain or arbitrary; fixed; >> established; definite. >> Thus indeterminate is the exact opposite of fine. You can't have it >> both ways. > Ooops. indeterminate is the exact opposite of finite. You've > uncovered a contradiction about the count of elements in an infinite > set that cannot be resolved from the definitions of finite and > indeterminate. karl m > Uh, no. Indeterminate just means unspecified, not infinite. Read the definition. Determinate=defined limit; indeterminate=undefined limit=infinite. That's why I'm using the 1913 version of Webster's, before modern mathematics took sway over the definitions. karl m === !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > Some finite, indeterminate number. You tell me the largest > finite number, and that's the set size. It doesn't exist? > Well, then, I can't help you. >> De*termi*nate (?), a. [L. determinatus, p. p. of determinare. See >> Determine.] >> 1. Having defined limits; not uncertain or arbitrary; fixed; >> established; definite. >> Thus indeterminate is the exact opposite of fine. You can't have it >> both ways. Ooops. indeterminate is the exact opposite of finite. You've > uncovered a contradiction about the count of elements in an > infinite set that cannot be resolved from the definitions of > finite and indeterminate. karl m >> Uh, no. Indeterminate just means unspecified, not infinite. > Read the definition. Determinate=defined limit; > indeterminate=undefined limit=infinite. Undefined limit is not the same as infinite. For example, in programming languages indeterminate loop forms are those for which you can't say in advance how often they will be run (i.e., while-loops, as opposed to the determinate for-loops). -- === > Some finite, indeterminate number. You tell me the largest > finite number, and that's the set size. It doesn't exist? > Well, then, I can't help you. >> De*termi*nate (?), a. [L. determinatus, p. p. of determinare. See >> Determine.] 1. Having defined limits; not uncertain or arbitrary; fixed; >> established; definite. >> Thus indeterminate is the exact opposite of fine. You can't have it >> both ways. Ooops. indeterminate is the exact opposite of finite. You've > uncovered a contradiction about the count of elements in an > infinite set that cannot be resolved from the definitions of > finite and indeterminate. karl m Uh, no. Indeterminate just means unspecified, not infinite. > Read the definition. Determinate=defined limit; > indeterminate=undefined limit=infinite. > Undefined limit is not the same as infinite. For example, in > programming languages indeterminate loop forms are those for which you > can't say in advance how often they will be run (i.e., while-loops, as > opposed to the determinate for-loops). Programming languages weren't invented in 1913. Please use analogies that are pre-Cantor. The opposite of undefined-limit is infinite. karl m === > Some finite, indeterminate number. You tell me the largest > finite number, and that's the set size. It doesn't exist? > Well, then, I can't help you. > De*termi*nate (?), a. [L. determinatus, p. p. of determinare. See >> Determine.] >> 1. Having defined limits; not uncertain or arbitrary; fixed; >> established; definite. > Thus indeterminate is the exact opposite of fine. You can't have it >> both ways. Ooops. indeterminate is the exact opposite of finite. You've > uncovered a contradiction about the count of elements in an > infinite set that cannot be resolved from the definitions of > finite and indeterminate. karl m >> Uh, no. Indeterminate just means unspecified, not infinite. Read the definition. Determinate=defined limit; > indeterminate=undefined limit=infinite. >> Undefined limit is not the same as infinite. For example, in >> programming languages indeterminate loop forms are those for which you >> can't say in advance how often they will be run (i.e., while-loops, as >> opposed to the determinate for-loops). > Programming languages weren't invented in 1913. Please use > analogies that are pre-Cantor. The opposite of undefined-limit is > infinite. Algorithms are definitely pre-Cantor. Euklid's algorithm (a few thousand years old) for finding the greatest common divisor of two numbers takes an indeterminate number of iterations. -- === > Some finite, indeterminate number. You tell me the largest > finite number, and that's the set size. It doesn't exist? > Well, then, I can't help you. >> De*termi*nate (?), a. [L. determinatus, p. p. of determinare. See >> Determine.] 1. Having defined limits; not uncertain or arbitrary; fixed; >> established; definite. >> Thus indeterminate is the exact opposite of fine. You can't have it >> both ways. Ooops. indeterminate is the exact opposite of finite. You've > uncovered a contradiction about the count of elements in an > infinite set that cannot be resolved from the definitions of > finite and indeterminate. karl m Uh, no. Indeterminate just means unspecified, not infinite. Read the definition. Determinate=defined limit; > indeterminate=undefined limit=infinite. Undefined limit is not the same as infinite. For example, in >> programming languages indeterminate loop forms are those for which you >> can't say in advance how often they will be run (i.e., while-loops, as >> opposed to the determinate for-loops). > Programming languages weren't invented in 1913. Please use > analogies that are pre-Cantor. The opposite of undefined-limit is > infinite. > Algorithms are definitely pre-Cantor. Euklid's algorithm (a few > thousand years old) for finding the greatest common divisor of two > numbers takes an indeterminate number of iterations. Ok, here's a definition of indeterminate from the 1913 dictionary: Indeterminate problem (Math.), a problem which admits of an infinite number of solutions, or one in which there are fewer imposed conditions than there are unknown or required results. karl m