mm-1829 I have not been trying to find a definition. I am perfectly happy with the standard one. My very first post in this thread was based on the assumption that number of elements can only mean cardinality, at which point I was accused of conflating the two concepts. Ever since then I have been challenging the doubters to come up with an actual definition of number of elements. No one has done so. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. You know, Dave, this is the second time you've said this so ignorance is no excuse. I've defined cardinality in universal terms as identical to number of elements and cardinality in parochial modern math terms as more akin to tangentiality than cardinality in universal terms. Lester Zick said: In other words, the inverse of the slope of the function defining the set is a measure of the size of infinity relative to the entire domain, right? -- Smiles, Tony Discussion, linux) You're slipping into utter incoherence. -- Jesse F. Hughes Dead men can't talk. Especially when they've been cremated. Jesse F. Hughes said: No I am not. Your insults only demonstrate your impatience and lack of attention. I have given some examples already. For instance, the integers map to the evens using the function f(x)=2*x, the derivative of which is 2, the inverse of which is 1/2 which is the number of evens per integer. In this simple case it works. For higher levels of function, which approach infinity faster, and have a larger derivative, the number of elements overall is smaller according to how quickly the function increases. Think about it for a little bit. It may come to you. -- Smiles, Tony Discussion, linux) These examples do nothing to define this notion. What is the function defining the set of all sets S c N such that N S is infinite? What is the inverse of its slope? This is incoherence and you pretend that these examples serve to create coherence. It might, but tomorrow is a work day. It wouldn't be wise to drink that much. -- Analysis/ editors have evaluated the paper, they accepted it for publication and they have the copyright of its contents - and thus they are responsible for its correctness,' she [said]. Jesse F. Hughes said: Is a set of sets defined by a function on reals? Can you take a derivative of a function on sets? Perhaps if you ask this 4 MORE times, the question will make sense. But probably not. -- Smiles, Tony Lester Zick says... Yes, you have said that, Lester. The problem is that that doesn't make a bit of sense to anyone except (maybe) you. -- Daryl McCullough Ithaca, NY Daryl McCullough said: It makes sense to me, as I've said. We'll get into that later, when you're ready...... -- Smiles, Tony On 22 Mar 2005 09:52:56 -0800, stevendaryl3016@yahoo.com (Daryl Well, the problem here, Daryl, is that mathematikers draw their definitions in very parochial rather than universal terms, i.e. sets have cardinality when . . . It's a very lame practice which I daresay is all too common among mathematikers. You don't like my defintion? Fine. Just try drawing a general definition for cardinality in other than set terms so that all of us may use it regardless of persuasion. Lester Zick says... On what basis do you call someone a mathematiker? Does that mean someone who is able to do mathematics? If you mean a professional mathematician, I'm certainly not one. Why should anyone care whether a definition is universal or not? I can't make any sense of it, one way or the other. Nor can I make any sense of your objections to standard mathematical definitions. -- Daryl McCullough Ithaca, NY On 22 Mar 2005 12:11:19 -0800, stevendaryl3016@yahoo.com (Daryl Solely on the basis of parochial mathematical definitions particularly in terms of set theory and cardinality defined in terms of set theory. Oh. Well I can't see any reason unless one wants to be universal with respect to ones knowledge. Clearly not yourself. Nor can anyone expect you to make sense of definitions which aren't drawn in universal terms to begin with. Now if mathematikers' definitions are only drawn in parochial terms, we can only expect parochial counterintuitive non universal results. How does that feel, Daryl? -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy I have asked repeatedly for a definition of number of elements and you have not provided one that is suitable for all sets. I have explained that tangents do not even apply to arbitrary sets. How do you take the tangent of the set of all bounded linear operators on Hilbert space, for example? The cardinality of a set is the least ordinal that can be mapped bijectively to the set. That is the universal definition in ZFC. It works for finite sets and for infinite sets. It doesn't matter whether they are sets of numbers or sets of functions or sets of prime ideals or sets of probability measures. The standard definition of cardinality is universal. Your mumblings are not. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Nor have you. And I have yet to see an arbitrary set. Probably the same way you assign mathematical properties to them. Except that this definitions assumes all kinds of non universal properties such as set, ordinal, mapping, and bijection. Which just means it's a parochial definition in universal terms. It works for no sets which aren't defined in universal terms to begin with but which mathematikers pretend are defined in terms universal enough for them to count, even if the count is uncountable. Get back to me, sport, when you can do mathematics higher than counting. What does tangentaility mean in this context? Bob Kolker As referring to derivative of content rather than the content itself. Don't you still have that dictionary that you advised me to use? -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy Nor an axomatization which endows the term with some meaning. There is no definition of point or line in synthetic geometry, but the axioms of geometry supply enough implied meaning that one can come up with models that satisfy the axioms of geometry. That is why we can draw diagrams. Bob Kolker cardinality Among mathematicians, perhaps. Call it I haven't seen those arguments. All I've seen are garbage cool consequences arguments and a conflation of methods to definitions. They should suppose I mean the common definition, not the mathematical one, unless the context is clearly mathematical. Is this too much to ask? If the definition is common, it should be easily accessible, right? I looked in a dictionary for number (though I really wanted number of). I got a couple of very interesting definitions: (a) devoid of emotion (b) indifferent Oh, wait -- the dictionary directed me to numb. Heh. Numb and Number. I then turned to number itself. There was not a common definition there were MANY definitions. Which one did YOU mean? The one that means singular or plural, maybe? [*] Personally I found most of the definitions to be reasonable summaries of how the term is used in common speech (that's what dictionaries are supposed to do) but not exactly models of scientific clarity. You know, the dictionary can define the aether, too. Among the definitions offered was a unit belonging to an abstract mathematical system and subject to specified laws of succession, addition, and multiplication; especially : NATURAL NUMBER This is from Miriam-Webster online -- is that common enough? In the interest of moving the discussion forward, could you agree that number of elements of a set X means that to the set X we will attach a number in the preceding sense? As a first step from that I might suggest we agree that the last three words cannot apply when discussing _infinite_ sets -- if you want to tell me that natural numbers can include infinite things then I would respond that the phrase natural number has no meaning at all, or at least that it means nothing at all different from what came in the other two lines of the definition. So perhaps you would agree that it's the two lines up to the semicolon that define what number means? dave [*] PS. I was going to write this in the style of Thurber but I can't bring myself to mess with the original, one of my favorites. This is the opening of a piece entitled, What do you mean it WAS brillig? I was sitting at my typewriter one afternoon several weeks ago, staring at a piece of blank white paper, when Della walked in. They are here with the reeves, she said. It did not surprise me that they were. With a colored woman like Della in the house it would not surprise me if they showed up with the toves. In Della's afternoon it is always brillig; she could outgrabe a mome rath on any wabe in the world. Only Lewis Carroll would have understood Della completely. I try hard enough. Let them wait a minute, I said. I got out the big Century Dictionary and put it on my lap and looked up reeve. It is an interesting word, like all of Della's words; I found out that there are four kinds of reeves. Are they here with strings of onions? I asked. Della said they were not. Are they here with enclosures or pens for cattle, poultry, or pigs; sheepfolds? Della said no sir. Are door Della said no again. Then they've got to be here, I said, with some females of the common European sandpiper. These scenes of ours take as much out of Della as they do out of me, but she is not a woman to be put down by a crazy man with a dictionary. They are here with the reeves for the windas, said Della with brave stubbornness. Then, of course, I understood what they were there with: they were there with the Christmas wreaths greatly relieved; we both laughed. Della and I never quite reach the breaking point; we just come close to it. [Copied faithfully but without permission from The Thurber Carnival, (c) through 1945 by James Thurber, published by Harper & Bros., pp. 43-46] Dave Rusin said: Very nice, Dave! Ain't words a wonderful diversion from numbers (and vice versa)? -- Smiles, Tony Discussion, linux) So far, your facility with numbers suggests a similarly bad facility with words. You seem to have trouble with definitions, context, forming well-formed assertions and so on. -- Jesse F. Hughes That's what's brutal about mathematics! When you're wrong, you can have spent years, and lots of effort, and come out at the end with nothing. -- James S. Harris on the path of self-discovery (?) Not quite the trouble you have, Dave. Jesse F. Hughes said: As opposed to all of the original thought that you've shared with the group? Gee, would that I were a creative genius like yourself, able to put together such whimsical poetic insults such as the masterpiece above. You are truly a marvel of genius, both analytical and literary! -- Smiles, Tony Did you check the dictionary for a definition of parochial? Discussion, linux) I repeat the motivation. ,---- | As far as I can tell, cardinality provides the best-motivated answer | to this question. | | Counting a finite set X involves creating a bijection between X and | some set {0,...,n-1}. In this case, X has n elements. | | Two sets X and Y have the same number of elements iff they both have | bijections to the same set {0,...,n-1}, equivalently iff there is a | bijection from X to Y. `---- But if you have another equivalence relation as well-motivated as the relation |X| = |Y| for an extension of set size to infinite sets, I'm all ears. I don't know any common definition of the number of elements of infinite sets. There's a difference between the meaningless use of undefined terms and common definition. If there *was* a common definition, I'd be happy to agree with your request, but it must be a real definition, not vague and ad hoc intuitions that N has twice as many elements as the set of evens and infinity - 6 more elements than {0,1,2,3,4,5}. Let's not mistake faux calculations for consequences from an actual definition. -- Puts his arm around you, fiddles with your hair. You know, and he says, come on, you know, just because you like a bit of a kiss and a cuddle with another man doesn't make you gay. Which, you know, I've thought a lot about. But I think it does. I think it does. --- The Office (interviews) from out your Except that they don't have to claim that, except in the sense of saying that your definition provides counter-intuitive notions and is not justified. Except that I never claimed that I refuted it, just that I saw no reason to accept the cardinality approach in light of the counter-intuitive consequences it results in. It isn't my fault you can't defend your own claims. Discussion, linux) That it provides counter-intuitive results on infinite sets is not a particularly negative feature. Our intuitions of the infinite are much less trustworthy than derived results from a well-motivated definition. That it is not justified as a notion of number of elements is false. It is an obvious extension of the equivalence relation generated by counting elements of finite sets. There *is* a clear philosophical argument for cardinality as a notion of set size. If you doubt that argument, then you need to do more than complain about its counter-intuitive consequences. That some fact regarding cardinality of infinite sets conflicts with your pre-theoretical notion about set sizes gives me no motivation to reject cardinality without an alternative definition. And, no, I have *still* seen no real alternative --- all I've seen is some ad hoc crap about if it's sets of numbers, then use order type and otherwise...? You have great faith in your intuitions about cardinality. Good for you. -- [I]n mathematics there are two types of integers: primes and composites. With such a distinction, it makes sense that there would be interest in them. It's like how in the world there are mostly two kinds of people: male and female [...] and lots of reasons for interest in the differences. JSH on math/biology it No, it's the proper answer. You and the other mathematicians are not looking for a definition, but are looking for a universal procedure for determining it, but this is not something that we can necessarily expect from any definition. For example, would you agree that we know what the definition of marriage is? Would you agree that there aren't any universal methods for determining if someone is married or single? And N, Q, and R are? What does it mean for the answer to be the best-motivated? What motivation are you aiming for? I don't think so, as I said elsewhere. It seems to me to be accidental that the bijection approach matches functionally what we do in counting. Clearly no one cares about a set of naturals when they are counting things, and clearly we can get the total number of things without doing that. And how do we know that this can be extended to all sets and still work? It seems to me that it should only work for finite sets, since it is only clear that number of elements and cardinality work out the same way for finite sets. Discussion, linux) I will pursue this particular subthread no further. You've shown that you haven't any real thoughts about what definition is at all. You are doing the reputation of philosophers no favors, I tell you what. (By the way, I haven't seen any journal-published philosophy of measure of set size. Do you know any?) The set of natural numbers, rational numbers and real numbers, respectively. Bizarre. Counting five rocks doesn't involve the elements of the set {1,2,3,4,5}? -- Jesse F. Hughes There are VERY FEW real mathematicians and I am one of them. Few of you can handle the pressure of real mathematics, like being wrong, while I demonstrably can. -- James S. Harris Although I have this thread in my killfile, I must have interrupted the filter process because I read that Since the notion of cardinality is not very differentiating in this context, people have adopted other notions of how big sets are. If cardinality doesn't capture the spirit of what you're trying to measure, you're perfectly welcome to invent other measures, as long as they are well defined. One of the ones people use specifically for subsets of N is called natural density and the number of elements in the set X which are no larger than n. There's no point in arguing whether cardinality or natural density or something else altogether is the right way to discuss how big sets are. You make a definition and then determine its consequences. Can you, for example, relate the sizes of sets X, Y, X union Y, and X intersect Y ? You've got to state what you think is true and then prove it. For natural density it's true that d(X union Y) = d(X) + d(Y) - d(X intersect Y), whereas for cardinality the best you can say is c(X union Y) + c(X intersect Y) = c(X) + c(Y), because - is not defined for cardinals; but each of these assertions is provable. I don't know what |N|/1000 means, but it is true that in (a) the natural density is 0. I have no idea what sqrt(|N|) means. The cardinality is the same as that of N itself; the natural density is zero. You're very glib about this f stuff so no one can tell what you really mean, but if you want f(n) to denote the n-th member of an infinite set of natural numbers, then in this case f(n) is more like n log(n). It is approximately true that S(n) ~ n/log n and so S(n)/n is roughly 1/log n, but that's much more information than you need to prove that the natural density is 0. I have no ideal what ln(|N|) means; it looks to me like you just happily write down names of functions and then write (|N|) after them. Picking up on the theme, you might want to consider some more complicated examples: (d) X = {1; 3; 9, 10, 11, ..., 30; 145, 146, ..., 840; ...} I'm choosing these numbers so that I have 100% of the numbers in a short range (from 1 to 1) 1/2 of the numbers in a longer range (from 1 to 2) 2/3 of the numbers in a longer range (from 1 to 3) 1/4 of the numbers in a longer range (from 1 to 8) 4/5 of the numbers in a longer range (from 1 to 30) 1/6 of the numbers in a longer range (from 1 to 144) 6/7 of the numbers in a longer range (from 1 to 840) and so on. The set is infinite. It has no natural density (that is, the limit I used in the definition does not exist). So how big would you say this set is? (FYI: The numbers in the last column of my table up there are of the form n! / (n-1). I know how I would compute the billionth element --- is that what you would call f(10^9) ? --- but I don't have a formula for it, and I have no idea what you're going to put in front of the (|N|) this time!) (PS: it's 7227020800 I think.) And where would you place (d) ? Before I nuked this thread I noticed non-mathematicians objecting (as is common) that mathematicians were getting persnickety about definitions as though that were the only way to approach the subject. It's not. It's perfectly reasonable to propose new ways to think about something, and these new ways can prove to be fruitful. But mathematicians know that appealing intuitive notions can lead to ambiguities which show that the proposed new ideas are incompletely formulated at best, and possibly contradictory. That may be fine for artists and philosophers, and some mathematicians count themselves in those groups; but in their professional capacity as mathematicians they know that this is a signal that the new ideas have to be reworked or dropped altogether. We think, for example, that the axioms of Zermelo and Frankl capture the notion of what a set is. Some day these axioms may be found to be inconsistent, and then we would have to stop using them. That doesn't mean that sets would cease to exist! Most mathematicians accept a Platonic notion that there are set-like things out there that we wish to use, and we would propose new axioms (definitions) which we think capture the central idea. But then we'd have to go through the same rigorous procedure of deducing the consequences of the axioms, seeing what they would say about bizarre situations, trying to break them, etc. As physicists do with gravity and as theologians do with God, mathematicians start with the intuitive notions like cardinality or set first, then try to make definitions which appear to embody what it is we want to talk about, and then use that language. If we get a _contradiction_, we MUST change our definitions since we reject the possibility that sets or cardinality don't exist; if we get a _paradox_ (unexpected result) then we MAY change our definitions and explore other alternatives, or we may accept that the unexpected result is real and revise our intuition accordingly. The definition of cardinality leads to conclusions you may find paradoxical; you are free to reject that definition while mathematicians have decided to keep it. BTW, I'm told that Banach and Tarski expected one reaction to their paradox and got the other instead. Well, you can reject (or use alternatives for) the Axiom of Choice, or you can accept it (and accept the consequences) -- both are valid mathematics. dave Dave Rusin said: Okay, at first I thought you were talking about the same thing I am, when you said natural density. Is the natural density of the integers zero, because they are an infinitesimal fraction of the reals? If so, then you need a unit infinitesimal so you can dsitinguish between subsets of the integers. I am trying to create a system that relates the levels of infinities explicitly and quantitatively, ultimately boiling down to levels of integration, as I see it. The idea is to look at the mapping functions and equate their inverses with the size of the resulting subset. For infinite sets a subset can be defined by the function without evaluating it, since the value will be infinite for an infinite set and infinite values. I have no idea how to evaluate this function offhand. My ideas are developing. How does one differentiate the size of this set from the size of other sets, in practice, or do they? Can you tell me whether this is larger or smaller than the primes? In a basket on the river, with a note........ Okay. This tends towards all of the numbers between 1 and oo, given every second term, so I would give it a measure of |X|, or perhaps slightly less. And yet mathematicians seem perfectly happy to ignore contradictions between their various treatements of infinite sets, or set realtions. Why is a proper subset of an infinite set not considered smaller than the superset? Why is bijection a valid measure of set size for infinite sets, when it fails to distinguish between sets hat obviously have these realtionships? These are the problems I am addressing here. Oh, good!! -- Smiles, Tony Boy, it's like you're a straight man, feeding me my lines. Mathematicians make definitions to match what it is they're trying to study. You want to study sets in the abstract? Fine -- we've got this notion of cardinality. It's real handy for distinguishing, say, sets that are small enough to fit inside the real number line from the really, really big sets. It's not so handy for distinguishing between, say, the set of primes and the set of composites. (They're both countably infinite, i.e. they have the same cardinality as N .) You want to study subsets of the real line? Fine again -- we've got a notion of measure that captures how much of the real line is included. It distinguishes between, say, [0,1] and [0,2] (which are of the same cardinality but not the same measure), though it still won't distinguish the primes from the composites (they both have measure zero). You want to study subsets of the natural numbers? We're ready for that too! We have this notion of natural density that probably matches what you're thinking of -- it gives a concrete way to describe the fact that there are many more composites than primes. Which one is the right way to measure bigger than? That's not a mathematician's job -- you go ahead and use the mathematical tool that best suits your purpose. Our job is to hone those tools and make sure we're certain what they measure, and exactly what you can depend on when you use them. So, see, your question above is perfect because it allows me to demonstrate how mathematicians can _clarify a question_ before being asked to give an answer. I don't understand any of that. I would probably not use the words larger or smaller at all because of the ambiguity. Each has elements that the other doesn't have, so you don't have either one larger in the sense of containment. You can put the sets into one-to-one correspondence so neither is larger in the sense that the set of left shoes in the world is as large as the set of right shoes (even though we have no idea how large either of the sets is, we know they're the same size, right? How?). The set of primes has natural density 0, which is as low as natural density can get, but I wouldn't really say the set of primes is smaller than X because X doesn't _have_ a natural density. Oh, and both sets have measure zero when viewed as subsets of the real line, so I have no reason to say either is larger in that sense either. So there is no definition I know of which allows me to say that one of these sets is larger than another. Is that a problem? I don't understand what that means. I even tried replacing the symbol between the two vertical bars with an N and I still could assign no mathematical meaning to this. I'll bet you can't either. No contradictions are known. Unexpected results, perhaps, but no statement P has been found which has the property that both P is true and P is false can be shown to follow from the definitions. Of course it's considered smaller-in-the-sense-of-containment. It might not be smaller by any of those other measures. So what? You know, at my institution, everyone has a GPA (average grade). Those whose GPA is sufficiently high can graduate with honors or with high honors or even with highest honors. Why is a person with a lower GPA not considered to have graduated with lower honors? Answer: official honorableness is not as refined a measure of student skill. Likewise we can have A properly contained in B but f(A)=f(B) where f is some order-preserving function such as cardinality, Lebesgue measure, or natural density. No one is claiming that these functions are one-to-one. I don't know what valid means. Do you mean logically consistent? No inconsistency of the axioms is known. Do you mean useful? Utility is measured according to the intended application. Do you mean legal? There are no laws prohibiting the use of any mathematical tool; More's the pity. People trained you to be an optimist, eh? I guess I was taught that, too, but recently heard a variant: When you're really angry with someone, remember that it takes 42 muscles to frown, but only 17 muscles to really beat the *** out of him. dave Dave Rusin said: Certainly! Nice clear description of the various approaches. I suppose what I seek is a system that ties them together cohesively, like agrand unifying theory of infinity hat demystifies it enough to make it usable. I am saying that if one defines the unit set size as the size of the naturals, that's an infinite unit of measure, so if you want your generalization to apply to finite sets, or infinitesimally infinite sets like the primes or squares, then you have to define infinitesimal portions of the unit infinity. I tend to think it is best to start with finite cardinality as defined, i.e. number of elements starting with 0 for the empty set, and define infinite sets as functions over the domain at hand. Then you can compare them according to those functions in terms of offset, ratio, relative slope, etc, in other words, based on derivatives of the function that defines the members, which would be inversely proportional to the size of the set. Certainly, for some functions it might not be possible to form derivatives that allow comparison with any other given function, but I think in many cases it's relatively straightforward. One can also look at the sum of inverses of an infinite series of numbers, and consider series that diverge more quickly to be larger, and series that converge to smaller values to be smaller. The convergence measure is straightforward. Is there a straighforward quantification of divergence for infinite sums? It would seem to need to be expressed as a function. Not as long as I am not expected to instantly come up with an answer when none of the extant approaches can. I still don't see how it can have a measure of zero when the billionth term of the series would state that of the first 7227020800 elements 1,000,000,000/1,000,000,001 of them are in the set. It seems like eventually the fraction not included is infinitesimal compared to N. Read above. Woudn't it be nice to have a cohesive system for classifying and comparing infinities and infinitesimals? I think the most important thing to keep in mind while doing this is that they are relative. For instance, 1 is infinite relative to 0, in a multiplicative sense, as oo is infinite relative to 1. Keeping this in mind, it may be best to define relative infinities rather than absolute infinities, so that certain dimensions of the infinity can be collapsed as needed. Wouldn't it be nice if they were in some sort of agreement? I picked up that signature from a coworker, a pagan lesbian drummer whom I like a lot, and who tends to smile. It's not my intention to piss anyone off, but to learn, teach, and make the world a better place by doing that. After all, our mind is our greatest asset. Good minds have good behaviors. Of course, sometimes people need their asses kicked. But then, of course, revenge is a dish best served cold. Anger is the fruit of hurt and hate is born of fear. Faith is lack of fear, and leads to hatelessness. That doesn't stop us from getting hurt and angry, but we certainly should not lose sight of the fact that when we are angry, it's because somehow we feel hurt. Maybe that's not the best time to act, when we're not thinking clearly. Maybe hurting someone isn't the best way to feel better. What goes around comes around, but does what comes around HAVE to go back around? Am I still in a philosophy thread? Oh okay...whew! -- Smiles, Tony interested person who simply happens not to have had much formal a nice fellow so I'll spend some time helping you out here, but I'm rediscovering now why I killfiled this thread and some of its participants, so I suggest we take this to email.) [It does have a (Lebesgue) measure of zero; but did you mean it has a natural density of zero? It doesn't -- it has NO natural density at all.] I think you've misunderstood what this set X of numbers is supposed to consist of. Or something -- your first sentence is numerically incorrect. What you should say is that of the first 7227020800 natural numbers, 1000000000/7227020800 of them are in the set. About one out of seven. But don't fuss too much over the specific numbers; you can figure out for yourself what's supposed to be in the set and why. Let's do this together. Here, do you know the game baseball? You've heard of Casey at the Bat? A great hitter, but moody: he lets the balls go by if he doesn't like them, but when he feels like it he swings and gets a hit. (Ignore the poem for now...) So imagine what happens to his batting average over time. He steps up to the plate; he swings; he connects! He's batting a thousand (1.000). You can't get any better than that. But then he lets a ball go by. That counts as a miss, so he's let his batting average fall to 1/2 (0.500). That's not a problem though: since he is a good batter, he can ALWAYS raise his batting average as high as he likes (except 1.000 is now impossible) just by getting enough hits in a row. So he hits ball #3 and raises his average to 2/3 (0.666). He can also obviously lower his batting average as low as he likes (to anything more than 0.000) just by letting balls fly by. So he doesn't even bother with balls #4, #5, #6, #7, #8 so that his average drops to 1/4 (0.250). And on and on he goes, changing his batting average from day to day depending on whether he's taken his daily steroids or not. One day he keeps connecting over and over until he raises his batting average higher than it's ever been since that first day. The next day he just lets all the balls go by, until his batting average is lower than it's ever been. As a result, we see his batting average rise to 4/5, then 6/7, then 8/9, etc., but in between those times, he lets his average fall to 1/6, then 1/8, then 1/10, etc. Now just write down the list of which balls he has hit: the first, the third, the ninth, etc. That's my list X. OK, so you're the baseball commissioner now, reviewing Casey's life's work. On your bulletin board you've got a picture of him at every at-bat. You see before you the long strings of hits, and the long strings of misses. The pivotal question is, do you want to say he has hit almost every ball? That he has missed almost every ball? Or what? In set-theoretic terms, this is precisely the question of how are you going to assign a measure of size to this set? I really don't care what you answer, because it's your job to decide whether to induct Casey into the Hall of Fame or into the Rogues' Gallery. You are certainly welcome to use any mathematical definitions or theorems you like to help you, and I can advise you what will happen if you try it. You can look at the set of hits. That's an infinite set, but in terms of cardinality it's not too big -- it's only countably infinite. There was the first ball he hit (ball #1), the second ball he hit (ball #3), the third one he hit, and so on. You can see what I've done: I'm establishing a one-to-one correspondence between the set of hits and the counting numbers N . That's what makes it a countably infinite set. You can also look at the set of misses. That's also a countably infinite set: there was the first ball he missed (ball #2), etc. Well, that's not very helpful is it? Any player with infinitely many at-bats is going to have the same verdict so far (countably infinite sets both of hits and of misses) except for the amazing sluggers who only missed a finite number balls, and the truly lame players who in their entire infinite career only connected a finite number of times. (That's me!) So you probably want a more refined way to measure the size of the sets of hits or misses, right? Now, what would you propose? A good choice would be batting average but how do you compute that? It doesn't help me to say divide the number of hits by the number of at-bats because I don't really know what the number of hits is here. That's an infinite set, remember. Here, I've got a calculator handy: what do I type in? Hmm. Mathematicians will simply be silent on the issue because they know that whatever definitions they try to make at this point (Like, say, When you need to divide two numbers and both of them are supposed to be the-number-of-elements-in-some-infinite-sum, just say the quotient is 1), those definitions cannot be made consistently with the usual rules for arithmetic (like the rule (a+b)/c = (a/c)+(b/c) ). We know it can't be done consistently, so we don't bother making up any rule at all. An alternative might be to say, watch the day-to-day batting averages and look at the overall progression. A person who hits every other ball starting with the first will have successive batting averages of 1.000, 0.500, 0.667, 0.500, 0.600, 0.500, 0.571, 0.500, ... and you can see that these are getting closer and closer to 0.500. So it's reasonable to say the lifetime average is 0.500 . That's exactly what the natural density measures; the natural density of the set {1, 3, 5, 7, ...} is 1/2. But our Casey doesn't HAVE a lifetime batting average because there's really no single number to which his daily averages are getting closer and closer. That's the price you pay sometimes in mathematics when you define something to be the unique thing with this property:... ; you're really stuck if there is no such thing (or if there's more than one such thing). So you see, there are some well-defined mathematical tools that you can bring to bear upon your problem. I don't start out with a pre-conceived notion of the right answer. In particular, I don't know whether I would call Casey a better batter or a worse one than the batter who hit every other ball. There is still plenty of room here for a new mechanism for ranking players' sets of hits, deciding when one such set is better (presumably meaning the same as larger here) than another one. Go ahead and propose some such measure and we can see what consequences it has. (It isn't really necessary that your measure of size be related to ordinary real numbers, the way a batting average is, though that would certainly be very comforting.) dave I meant 1/1000, sorry. Dave Rusin said: OH!! Well then that makes some sense. Okay. And |N| would be 1 then? And the set of primes would tend toward an infinitesimal fraction of the integers and have overall desnity 0? That makes some sense, but when lots of different size infinities all end looking like zero, there more refinement to be done I think. -- Smiles, Tony Oh goody, then I'm +1 as a teacher, today. You're using |X| to mean natural density then? I didn't, and no one else does, but just for now I'll use your notation; then yes, |N|=1. By the way the natural density of the set of all composites is also 1, so this notion of natural density isn't going to completely square with your intuition that our measurement of size will report something strictly smaller for proper subsets. Just so you're forewarned. If you skip all the words from tend toward... to ...and, then yes, that's fine. The set of primes is a set (of natural numbers); to each set of natural numbers I've assigned a number (between 0 and 1 inclusive), which means the density IS something (0 or 1/1000 or whatever) -- it doesn't TEND TOWARD anything. You look at the set, do some private computations, and then announce the label you're going to put on the set. In this case, the label is zero. That is, the density is zero. You could also look at the _sequence_ of numbers S(n)/n which I mentioned last time; it makes sense to talks about a sequence tending toward something. In this particular case S(n) is the (GASP!) Prime Counting Function, so the 100th term is S(100)/100 = 25/100 = 1/4, and the fractions generally decrease from there, tending towards zero. I don't think infinitesimal fraction means anything. There are no infinitesimal rational numbers, anyway, and all the S(n)/n 's are rational numbers. [Integrity alert: I am also glossing over the fact that d(X) is defined as a limit, and sometimes the correct value of a limit is Does Not Exist. So when I say every set of natural numbers has a natural density, I mean that for each X there is either a number _or the symbol DNE_ which is attached to the set X. I didn't want to detract from the points above, but I also don't want to miss this issue. I notice you didn't have any response to the other set I described last time -- one whose natural density really is DNE.] By the way, you have a typo here: You mean overall destiny 0, right? Like the best-laid plans that gang aft agley? Well, that's fair. Like I said last time, you start with an intuition and make a definition that you think gets to the main idea. There's no guarantee that it will do what you want it to do, but if it doesn't you can attempt to refine your definitions (what do you propose?) or learn to live with the consequences. (The rest of us do the latter: we accept the fact that there is not a bijection [OH NO!] between the subsets of N and the possible densities in [0,1]). You know, I could really roil the waters here and mention that natural density is taken as a stand-in for the concept of probability, as in When you pick a number at random, what is the probability that it is prime?. But no, I won't mention that. dave Dave Rusin said: Very well, then, I have prepared myself, Continue.....(braces himself) I meant that as n increases, the fraction of integers less than n that are primes becomes a smaller fraction of n, tending toward an infinitesimal fraction at n=oo. Hence the measure of zero. And yet, the set is still infinite....hmmm..... Seems zero should refer to the empty set.. infinitesimal fraction=zero if you like. There are infinitesimal number systems, apparently. Maybe they'll be of some use..... Actually, I noted that as the intervals from 1 to n get longer, the proportion of the set at every other iteration gets closer to 1, so that as n reached infinity, the proportion of the naturals NOT included would be infinitesimal compared to N. The natural density is really 1 in this case, as far as I can tell. No, I meant the desnity, as in the desnitatious desnitification, due to the intrinsic presdesnitinance inherent in the process of dedesniticulation. Duh! Oh wait.... No, you're right, I meant density. Ooops. My bad. :} Yes, that makes perfect sense and their may be discoveries in probability that can be applied. I am leaning more toward calculus and sums of infinite series, personally. -- Smiles, Tony Jesse F. Hughes said: This is what I get for not being online for the weekend. It's nice to see the subject is at least worth arguing about, even if no one gets anywhere. LOL! Okay, now I am back at work (slaving). I have already posted replies about an alternative, though obviously I don't have a perfect alternative, or you'd be reading my paper in a math journal. :) -- Smiles, Tony Dave Seaman said: The reordering of the numerical elements of a set from their natural quantitative order for the purpose of deriving a bijection between TWO sets is what causes the unsound conclusion, if a bijection is achieved, that the two sets have the same size, because they have the same cardinality, as it is defined. If you can draw a bijection without violating the inherent quantitative order of the numerical elements, then you can demonstrate a real equivalence in size, but applying functions to numerical elements that no longer have their original meaning because the order that defined it has been changed introduces contradictions that result in unsound equivalences. How do you explain that there are the same number of integers as rationals, without resorting to an artifical reordering of the numbers and relying on the proof of bijection? You can't. In fact, regular good old-fashioned logic and a quick peek at the number line says otherwise. Why should I trust a complex method with undemonstrated validity over regular good old-fashioned unbreakable basic logic? -- Smiles, Tony I asked for your definition of number of elements. Instead of answering the question, you simply resorted to providing yet another laundry list of reasons why you find the notion of cardinality objectionable. Are you ready to concede yet that you do not have a definition of number of elements? By the way, mathematicians don't claim to have a definition for that term, either. What they have is a definition of cardinality. In informal speech among mathematicians, it is commonly understood that number of elements is to be interpreted as cardinality, not because mathematicians claim the right to dictate the English language, but simply by default, on the grounds that no alternative definition of that term has been proposed. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Well this is true but not really germane to whether cardinality already has another, universal, definition in ordinary mathematics apart from set theory. Simply noting that neither modern math nor ordinary mathematics has a decent definition for number of elements doesn't mean the modern math definition of cardinality is correct. Nearly all of mathematics is built on set theory as a foundation. Cardinality is a concept that applies to sets, and its definition comes from set theory. And there is nothing particularly modern about set theory or cardinality, which have their roots in the 19th century. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Not even sparsely true. No, no, the definition comes from universal concepts of cardinality having nothing to do set theory. Modern mathematikers like to pretend the world began with set theory but that's only because they're too lame to think for themselves. Well it's certainly good to know that things having their roots in the ancient past of the 19th century are not particularly modern when much of ordinary mathematics dates to the more recent past of the 17th-18th centuries and even more recent past of the 25 or so centuries BC. Lester Zick said: I would say that the set of mathematics built on set theory to the set of all mathematics is as the set of primes to the set of naturals. Is that what you're saying? Now now be nice. Sometimes people have to be forced to think for themselves! LOL!!!! Touchee! -- Smiles, Tony I would like to know which areas of mathematics you think are not built on set theory. It can be argued that category theory is not necessarily based on set theory, since category theory can be used as an alternate foundation in place of set theory. I can't think of many other parts of mathematics that are not conventionally based on set theory. In nearly every case, we define things by starting with sets and then adding some structure. Take the definition of the real numbers, for example. Real numbers are sets. (They are usually defined either as Dedekind cuts or as equivalence classes of Cauchy sequences, but in either case, those objects are sets.) Lots of things are defined as ordered pairs or ordered triples of sets. A topological space is a pair (X,T), where X is a set and T is a subset of P(X) satisfying the definition of a topology. A measure space is a triple (X,S,m) where X is a set, S is a sigma-algebra on X, and m is a measure defined on S. And so on. And by the way, ordered pairs and ordered triples are also sets. So what exceptions do you have in mind? -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Oh, the definition of circles for one. See, Dave, this is a real problem with mathematikers' arguments, such expressions as it can be argued that . . . when universal knowledge is couched more in terms such as it is that . . . You have the very bad habit of saying such things and then assigning the origin of such expressions to those who criticize you for making them. Is this a joke? You define things any way you want and then claim they're mathematical definitions. Good luck with that, Dave. The last I heard mathematikers were pretending all reals fall on a real number line. A Dedekind cut is the unkindest cut of all. Yes, yes, this clears everything up all right. More along the lines of least ordinals defined as sets. And what would that be? -- Giuseppe Oblomov Bilotta Can't you see It all makes perfect sense Expressed in dollar and cents Pounds shillings and pence (Roger Waters) Dave Seaman said: Sorry. Another glib remark. My comment was basically meant to imply that it may not be so easy to quantify exactly how much of math is set theory-related. Is this thing on? I guess this is why there are no mathematical comics... :( Simply the set of all things which are not sets. Duh!!! ;) -- Smiles, Tony You have no idea what you are talking about. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Yes. Fortunately I have a much better idea what you are talking about. Dave Seaman says... Exactly. Lester has no earthly idea what he is talking about. But he (in a tag team effort with Allan, Albert and Tony) have managed to keep a thread going for a thousand posts based on a topic that he has no understanding of, and no actual interest in. It's an amazing accomplishment in its own right. -- Daryl McCullough Ithaca, NY Daryl McCullough said: Come on. You've enjoyed every minute of it. Have you ever fought so hard to defend Cantor? Isn't it good exercise? agreed that it could not be agreed on whether the answer was aleph_1 or c, if there is really any difference between them, which I guess is still an open question, eh? Fodder for the fiddles...... -- Smiles, Tony Tony says... Yes, you're right. It is sort of fun to be challenged to defend an idea against brand new objections. It sometimes does bring about a new understanding of the concepts involved. -- Daryl McCullough Ithaca, NY Daryl McCullough said: Does it maybe seem to be doing a little of that? Maybe in time....it's good exercise in the meantime, for everyone, I think. Do you really think these are new objections? -- Smiles, Tony On 22 Mar 2005 09:57:26 -0800, stevendaryl3016@yahoo.com (Daryl Yes, and I certainly appreciate your acknowledgement, Daryl. Now if we could just get back to the subject at hand which was universal knowledge and not the interminable whinings of mathematiker hacks. Not so amazing, given the knee-jerk reactions of mathematikers when someone forces them to actually examine their presumptions. -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy I would classify mathematics from the time of Galois onward as modern. Bob Kolker And I would classify mathematics from the time of Bob on as parochial. Dave Seaman said: equivalence between sets: An ordered bijection, as you had suggested, works without warping the relationship between the sets. A reiteration of the contradictions between cardinality and more basic logic seemed in order, since it seems so hard to get across. Bijection aside, wouldn't you judge that there are more rationals than integers, looking at the number line? -- Smiles, Tony You have not demonstrated any contradictions. There is a difference between a contradiction and something that is merely counterintuitive. You are talking about order isomorphism, which I have already defined elsethread. So let's apply your proposed definition. Which of these sets has more elements? (a) The reals in [1,2], (b) The reals in (0,3). Notice that there is an order isomorphism between (a) and a subset of (b), and there is also an order isomorphism between (b) and a subset of (a), but there is no order isomorphism between (a) and (b). So your definition does not seem to yield a total order. This can't happen with cardinality, by the way. It's a theorem that if A and B are sets such that there is an injection from A into B, and also an injection from B into A, then there is a bijection between A and B. No, I would conclude that the rationals have a higher density than the integers, looking at the number line. Cardinality doesn't look at the number line, because by doing so you are limiting your attention to subsets of the reals, and cardinality applies to all sets. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Dave Seaman said: Well, obviously I am not talking specifically about an order isomorphism, then, if the addition of an endpoint is considered all that important. I would characterize the relationship this way: Map all but the endpoints of [1,2] to (0,3) using the formula f(x)=3*x-3. Therefore the second set has 3 times as many elements, offset by 3. Finally, subtract the two endpoints. If the size of the first is X, the size of the second is 3*(x-2)=3*x-6. How quantitative!!! Hmmm...Now the density is greater for rationals.....is the number line shorter for rationals? If you have the same space with a higher density of elements packed in that space, then you have a greater number of elements in the denser space by the definition of density. At least, that's the definition of density in every other area. -- Smiles, Tony Discussion, linux) And how does it compare to the power set of N? We are talking about arbitrary sets, no? -- Jesse F. Hughes Of course, my ability to admit my mistakes and correct them is a trait that many of you seem to never have properly appreciated. -- JSH, discussing his 1463rd proof of Fermat's Last Theorem. And the definition of arbitrary sets would be? Jesse F. Hughes said: It is the derivative of the power set of N. -- Smiles, Tony Discussion, linux) Er, since power set of N is a constant, I guess that the size of N -- Jesse F. Hughes And I'm one of my own biggest skeptics as I had *YEARS* of wrong ideas, and attempts that failed. Worse, for some of them it took *MONTHS* before I figured out where I screwed up. -- James Harris Jesse F. Hughes said: Why no, N is an infinite function, of course, in Bigulosity Theory. And I thought I was clear about that! ;) It was glib answer, but derivatives are going to play a part in Bigulosity Theory, in conjunction with LHospital, and sums of infinite series, to form a cohesive picture....... -- Smiles, Tony Discussion, linux) Yeah. Good luck with that. Really. You're sounding less coherent every post. Honest. It's one thing to complain that cardinality doesn't match your intuitions. It's another to talk about derivatives of arbitrary sets. The former is simpleminded arrogance and the latter is nuttery. -- Jesse F. Hughes Leaving things always seems to fix me, Running seems to ease my worried mind. -- Bad Livers, Honey, I've Found a Brand New Way Jesse F. Hughes said: I have already explained that it's not just my intuitions that it contradicts, but other mathematical ways of evaluating sets. Did you forget that part? The derivative functions are for distinguishing subsets within an arbitrary set, not for distinguishing arbitrary sets from each other. So, I am incoherent, arrogant and nutty? I see. Well, that's better than being uncreative, inattentive and dense, so I guess I won't complain. Good luck with that. -- Smiles, Tony Of a person, you can measure the height, the weight, the number of molecules that form its body, its age, etc. Does each of this datum contradict the others? No, they are simply unrelated. Likewise, cardinality (how many elements are in a set), Lebesgue measure ( b-a being the Lebesgue measure of the interval [a,b]), density etc are totally different and only loosely related concepts. They may refer to the same set, but they do not contradict each other. -- Giuseppe Oblomov Bilotta Can't you see It all makes perfect sense Expressed in dollar and cents Pounds shillings and pence (Roger Waters) Discussion, linux) Yes, fine. Let our arbitrary set be the powerset of N. Let X = { S c N | N S is infinite } Let Y = { S c N | S c Primes } That is, S is in X iff the set of natural numbers *not* in S is infinite and S is in Y iff every number in S is prime. Tell me about the derivatives of those two sets. I can't seem to calculate them. (I *can* calculate their cardinality, on the other hand.) -- Jesse F. Hughes But nothing's being Dr. Ullrich is a particular case of something's being such that nothing is it: (Ex)~(Ey)(y = x) -- John Correy on the failings of first order logic Jesse F. Hughes said: Gee, lemme guess.....aleph_0? Wow! How'd I come up with that? (sigh) Perhaps you can find some more sets of sets defined by non-differentiable functions....go for it. -- Smiles, Tony Discussion, linux) Here is my function for the set of even numbers. Pick some rapidly growing function A (like, say, the Ackermann function). Define f(n) = 2m where m is the unique number such that A(m) < n <= A(m+1). Now, you offered a different function for the same set. You offered g(n) = 2n. Which one am I supposed to differentiate[1] in order to determine sizes? f grows remarkably slower than g, so the set defined by f must be much smaller than the set defined by g, right? Footnotes: differentiation or tangent lines mean? -- Eventually the truth will come out, and you know what I'll do then? Probably go to the beach. I'll also hang out in some bars. Yup, I'll definitely hang out in some bars, preferably near a beach. -- JSH on the rewards of winning a mathematical revolution Discussion, linux) Wrong. Look, do they distinguish subsets of an arbitrary set or not? -- And God Himself won't help you if this goes bad as despite your beliefs I can assure you that angry people will against all law if necessary tear their rage out of your hides if it goes badly. -- James S. Harris, on the dangers of criticizing his mathematics Please define the above sentance. What do you mean by it? Bob Kolker robert j. kolker said: I dunno . What was the question again? I think that was a glib stab in the dark kind of answer, but it's probably correct anyway. Say, what my first response to this thread when you started it? Was that the right answer according to cardinality, or was it c, or was that not decidable? Oh yeah, I think that depended on whether you believe in the axiom of choice or something. -- Smiles, Tony What do you mean by It is the derivative of the power set of N Bob Kolker Or you could just begin, Bob, by defining the above sentence and what you mean by it, you know, just to get us started. Not in every other area. Only in the areas that are limited to finite sets. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Jesse F. Hughes said: If this were simply a matter of vernacular usage and colloquial equivalence, then why would mathematicians get upset at the notion that cardinality is not a very accurate description of the size of an infinite set, and why do they defend their right to draw the conclusion that there are no more rationals than integers, and no more integers than odds, based on cardinality measure? This is no meer choice of words, but an inappropriate drawing of conclusions based on a system which, while brilliant, lovely, rigorous and largely internally largely because people have a hard time refuting it empirically, and yet, it contradicts conclusions drawn through other means, which is about as empirical as one gets in mathematics of infinity. Is it so hard to admit that cardinality is not the last word on infinity? -- Smiles, Tony [. . .] Tony, let me suggest an alternative view of what is going on in modern math and set theory where infinities are concerned. When mathematikers use the term cardinality, they aren't talking about the same concept used by mathematics in universal terms.They're talking about tangents. Let's say we're discussing a set with a finite number of elements. Then the parochial modern math definition of cardinality coincides with the universal definition of cardinality. But when the number of elements is undefined or infinite there is no way to compare the sets directly. In ordinary mathematics we then resort to L'Hospital's Rule to compare the sets in terms of tangents. For straight lines, however, the cardinality of the set and the slope or tangent of the line are identical. And finite sets of points are all straight line segments. In other words, set theory simply takes cardinality to represent the slope or tangent to straight lines or curves instead of the number of elements. The difficulty is that modern mathematikers probably have no idea what they're actually talking about. They use the term cardinal as if it were the number of elements when discussing finite sets then go on to use it in parochial terms for infinite sets.They just confuse tangents to lines with the lines themselves. This was probably done deliberately originally but nowadays I expect this confusion is just part of history and probably goes unrecognized by mathematikers. Lester Zick said: Yes, as I said before, I believe that L'Hospital will become central in the refinement of size measure for infinite sets, as the slope of the function defining members of the set that approach infinity will be inversely proportional to the size of that infinity, or density of saturation of the real numbers. These ideas need to be integrated better before I am comfortable with the results they imply. I see no way to reconcile the idea that integers are equinumerous with rationals, despite equal cardinality ala Cantor. Again, I thank you for reminding me of L'Hospital. :) -- Smiles, Tony Wow. He actually said this? How helpful of LZ to tell the world what we in ordinary mathematics do. I guess I must be an extra-ordinary mathematician then because in my world, (a) number of elements is undefined or infinite suggests a synonymy which I never hear among mathematicians (b) L'Hopital's Rule, overused as it is in the Calculus, has nothing to do with figuring out the size of sets (c) To say that L'Hopital's rule is related to tangents is a tremendous stretch of analogy (d) I cannot see any connection between tangents and cardinality either. I won't go on -- the mental gymnastics in those two sentences is already enough to wear me out. What a tour de force -- congratulations, Lester! Ah, so you're an acolyte in the worship of nonsense then? Let me advise you not to waste your type with LZ -- go straight to Alan Sokal's masterpiece. dave Dave Rusin said: Dave - Lester's talk may sound like a bucket of snakes most of the time, but i am convinced that he has valid concepts in his head that he is trying to express. It may take some time to make sense of what he is saying, but if you read between the lines like myself, you too may find yourself saying, ohhhhhh that's what you're saying!!. Lester's a good fellow with a funny sense of humor and way of winding words that wrinkles the cortex just a little bit extra. In my response to that post of his, I may have clarified what he is saying, because it's not as vacuous as it sounds. He's a smart fellow, just not the clearest communicator. -- Smiles, Tony Discussion, linux) Your convictions that the nonsense regarding L'Hospital do nothing for the plausibility of your other judgments. -- If you have a really big idea, you can get a measure of how big it is REALLY, REALLY, *REALLY*, BIG DISCOVERY!!! --James Harris, on being ignored Jesse F. Hughes said: Nor should they detract from it before you have any idea what I am referring to. Just because the connection is not immediately obvious to you doesn't mean it's not valid. Maybe you have just never made that connection before. I'll do some thinking on it before I prersent any arguments in favor. Suffice it to say for now that in my opinion, Lester's references to tangent and L'Hospital are not far off the mark, despite his sometimes awkward style of aiming. -- Smiles, Tony Discussion, linux) ^ are sensible -- So, at this time, I'd like to assure you that I am not interested in making sure mathematicians worldwide get fired. I've rethought my desire to go to Congress and try to get funding for mathematicians cut. -- James Harris is a reasonable man. Whew! We in ordinary mathematics or we in modern math? Mathematikers hardly ever hear things. No, but matching counted uncoubtable countables seems to. Or an exactly apt description. Only in set theory. Tell me, Dave, do you and your mental gymnastic teammates play handball against a curb? And what a shame you don't take your own advice but propose to counsel others anyway on whose advice to take. as far as I can tell, are not really related to mathematics or to sci.math (where I am reading this), except for one which I cannot manage to leave without a reply: It's an apt description of WHAT? Could you kindly state LHopital's rule? Just FYI, the One True Source Of Earthly Wisdom, which is Google, finds that only 5-10% of all web pages which match LHopital or any variant will also match tangents, and the matches are, to my way of thinking, essentially accidental. Hardly an exactly apt description. So, again, do you know what L'Hopital's rule says? dave L'ook it up, D'ave. Yes. Albert Wagner said: Well, Albert, much as I appreciate your support, I think we're coming from different places. I don't just simply not like the definitions. I have a specific problem with conclusions of sameness derived from these definitions, where I see a clear difference according to other definitions. Cardinality may be internally consistent (except for the exceptions regarding infinite sets), but it is not consistent with some other ways of viewing sets. It seems like we need a non-euclidean synthesis of these methods to bring them into more general agreement. Unlike yourself, I see much more potential than we've realized for dealing with infinities and zeroes, and would like to see things progress a bit more in a more consistent manner. It's the Jnani in me. ;) -- Smiles, Tony Well, I didn't really have you in mind when I made my remark. My use of 'we' rather than 'I' has caused much confusion among many posters from sci.math concerning my identity and position. I regret it. I had hoped by that post to extricate myself from the noise surrounding discussions of mathematical definitions. I find them mildly irritating but have no problem with mathematikers speaking whatever language they like when in a group of their peers. My earnest hope is that they, and all others who delight in talking of infinities as if they had anything to do with reality, such as yourself, would remove philosophy NGs from their crosspost lists. -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy Discussion, linux) Hey! We agree on this. Some of the questions discussed here have both mathematical and philosophical content, but I sure don't know why they're crossposted to newsgroups about A.I. and meta-philosophical issues[1]. They seem bloody off-topic there. My apologies for not removing the excessive crossposting, but I expect that if I did, then the correspondents would not see responses. I think that the folks posting from those groups should be responsible for determining what's on-topic there and I would be happy to obey followup-to headers[2]. Footnotes: [1] I assume that's what sci.philosophy.meta means. [2] As long as they include sci.math, presuming that there is still mathematical content. -- When you go to class today, if your professor talks about algebraic number theory, or misuses Galois Theory[,] I want you to carefully notice how you feel. Hold on to that feeling so that you never forget it. --James S. Harris, on channeling rage via Galois theory. Albert Wagner said: Well, Albert, there are certainly threads here in which I do not participate, since they don't interest me so much, or I don't have enough background to form an opinion. We don't have to participate in EVERY discussion. Is that a royal we? ;) -- Smiles, Tony Dave Seaman said: I think the normal definition is good. Count the elements. When the set becomes ifinite that's impossible, so cardinality was invented to virutally enumerate the elements using mapping functions between sets of numerical elements. When using numerical elements, such as integers, rationals or reals, each number type saturates the real number line to a different extent. Personally, I see the rationals as being more similar to the reals that integers in this respect, since they fully saturate the line from a finite perspective (though perhaps porously), which integers do not. In any case, when you are looking at infinite sets of numbers in finite sections of the number line that have the same saturation on the number line, such as [0,1] and [2,4] in the reals, the relative sizes of the sets should be considered to be the same as their relative covered sections of the number line. I'd say the size of the set should be the product of its saturation and domain on the number line. The domain may be easy to measure. The saturation is a numerical representation of something like order type, and needs to be formally defined. -- Smiles, Tony The rationals are dense in the reals. But you are talking about properties of ordered sets, not properties of sets. Elsewhere I have explained the difference between a bijection and an order-isomorphism. If you want to talk about the order type of a set, fine. But the order type is not a particularly good indication of the size of a set. Consider the sets w and w+1, for example: w = { 0, 1, 2, 3, ... }, w+1 = { 0, 1, 2, 3, ..., w }. These sets have different order types. In particular, w+1 has a last element, while w does not. But both sets have the same cardinality. Can you arrange these sets in order of number of elements? (a) the set of of multiples of 1000, (b) the set of squares, (c) the set of primes. All three of those sets have the same cardinality. Which has the most elements, according to you? Perhaps you can explain what your definition says about this situation. Here's another example. Which of these sets has the most elements? (a) The set of rational numbers. Notice that (a) is everywhere dense, while (b) is nowhere dense. Thus it seems that (a) should have more elements according to you. However, (a) is countable and (b) is uncountable. Thus, (b) has the larger cardinality. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Dave Seaman said: Well, from what little I gleaned of order type in quick lookups, I sisn't get the impression that it really quantifies things any more exactly than cardinality, though I probably missed pertinent applications or extensions of it. Whether something has a last element, despite being infinite, doesn't necessarily seem to be of usable consequence at first glance. hard problem but... Therefore |c|<|b|<|a| in my book. I'd have to think about this one a bit. The Cantor set does seem to be sparse, so by my thinking, over a given interval, it should contain a smaller infinity than the rationals. The problem here may be in the construction of the Cantor set in a way that doesn't reflect the quantitative order of its members. This makes me wonder about the validity of my apporach to enumerating numbers from 0 to infinity this weekend by similar generations of in-between values to saturate the numbers, but I guess I'll see where it takes me. Possibly into a black hole. :) -- Smiles, Tony Discussion, linux) What is the definition of |x| that supports these calculations? -- Jesse F. Hughes I often told you of the dangers of hubris, and most importantly of all, I TOLD you that I wanted to change the institution of mathematics worldwide. -- James Harris, on the evils of pride Jesse F. Hughes said: That notation is supposed to denote the cardinality of the set x. -- Smiles, Tony Discussion, linux) Obviously not. Cardinality is well-defined and supports none of those claims. In terms of cardinality, it is utterly, stupefyingly clear that |a| = |b| = |c|. Dave had given you an out by using the vague term number of elements, but your response says that you meant the well-defined term cardinality. Clearly, then, your answers are just wrong. (There is also no definition of division, square root or logs of arbitrary cardinalities, so your answers are worse than wrong. The calculations that lead to the inequalities are literally meaningless.) -- When you go to class today, if your professor talks about algebraic number theory, or misuses Galois Theory[,] I want you to carefully notice how you feel. Hold on to that feeling so that you never forget it. --James S. Harris, on channeling rage via Galois theory. Jesse F. Hughes said: Doh! (bangs head on rock) NOT cardinality. oops! Size of the set by my thinking....let's call it bigulosity. The bigulosity of the set X shall henceforth be denoted by !@#(X). What I meant to say was: !@#(c) < !@#(b) < !@#(a) Ah, but that is not true of bigulosities. In bigulosities one can define relative levels of infinity using mapping functions and derivatives thereof, for a fine granularity of values. :) -- Smiles, Tony Discussion, linux) Wow! Sounds great! I eagerly await a sketch of the theory! -- Jesse F. Hughes Well, talk to her. Tell her about your feelings in an open and honest way. Yeah. Either that or be a man. -- Futurama Jesse F. Hughes said: It's all in the marketing. But don't answer yet, because for just three easy payments of 19.99, you also get the handy dandy....... -- Smiles, Tony The Cantor set is the set of all real numbers in [0,1] having a base-3 expansion in which only the digits 0 and 2 appear. How about these two sets? (a) The reals in [1,2]. (b) The rationals in [2,4]. Since (a) is dense in [1,2] and (b) is dense in [2,4], an interval twice as long, are we to conclude that (b) has more elements? Notice that if you actually had a definition of number of elements, it wouldn't be necessary for me to ask questions like these. Such questions could be answered in principle simply by applying the definition (although the actual application of a definition is sometimes problematic. Consider the definition of a nonempty set. Simple, right? So, is the set of odd perfect numbers nonempty?) -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Dave Seaman said: Despite the equivalence of the sets of integers and rationals according to cardinality, rationals and reals seem more similar. Exactly what is the difference between the reals and all numbers representable by decimal numbers of an arbitrary number of digits? Can't those digits be translated into a rational number? What real numbers are there that aren't rational? Is this a dense set? If more than half the reals are non-rational, then the first set would be larger, otherwise the second set would be larger. Countability and bijection aside, I am not sure there isn't ultimately an equivalence between the rationals and reals, except perhaps for a very tiny minority of transcendental numbers or other irrationals. After all, couldn't you find a rational number between any two real numbers that were well defined? Seems like it to me. You have already admitted that you don't have a definition for number of elements, so it seems unfair to expect that of me. What you have is a method called bijection that you use to calculate a measure of set size called cardinality, and a colloquial understanding that this is how we measure the size of sets for infinite sets. I am suggesting some improvements to the method, but don't yet have a fully fleshed out alternative. -- Smiles, Tony That was a trick question. Almost all of the real numbers are irrational. That includes numbers like pi, e, and the square root of 2. There are only countably many rational numbers, and therefore the irrationals are uncountable. It's a lot more than half. It's almost all, in the sense that the rationals have Lebesgue measure zero. But I'm not the one claiming that number of elements must mean something different than cardinality. If you claim the meanings are different, then you must have meanings for both those terms in mind. What are the meanings? -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. If in the arbitrary you include also infinte number of digits, then the difference is none. Sometimes yes, sometimes not. For example, the square root of two. Yes. Density has nothing to do with cardinality (size of the sets). In layman terms, density is a way to describe how a particular set is distributed wrt to the other. Trascendental numbers and other irrationals are not a minority. Indeed, they are the majority of real numbers. The reason why they are less known is that humans tend to think in terms of natural numbers and their ratio. Yes. This is exactly why rationals are dense in the reals' set. See how density describes distributions of sets within other sets? -- Giuseppe Oblomov Bilotta Can't you see It all makes perfect sense Expressed in dollar and cents Pounds shillings and pence (Roger Waters) Giuseppe Bilotta said: But, given any decimal fraction of n digits, it can be expressed as those n digits as an integer divided by 10^n, a rational number. So, why can't any decimal fraction be expressed as a rational? But, if you can calculate it to an arbitrary number of digits as a decimal number, and then translate that into a rational, what's the difference? Remember, we're dealing with infinite sets, so you can't say we can never use an infinite number of digits, and in the same breath say you can extend some mapping infinitely and declare it complete, when it is never complete or even in any finite number of iterations. Assuming we have an infinite number of decimal digits, why can't we calculate root 2 and turn it into a giaganti fraction? Is it still? Perhaps that is true. Is there proof of the realtive numbers of rationals and irrationals among the reals, bijection aside? And you can find a real between any two rationals? Waitiaminit, this sounds like complementary injections. Doesn't that constitute a bijection for the purposes of cardinality? Are rationals really any less numerous than reals, or are they just more enumerable? -- Smiles, Tony Discussion, linux) Did you know that real philosophers of mathematics take the time to learn basic math before overthrowing ZFC as unintuitive mish-mush? Just thought I'd mention it as an option. -- If you see math knowledge as a tool--as a hammer--with which you can attack other people then ... you defeat rational discourse. I get to call my proof the Hammer. It's more powerful than *any* physical object. It is overwhelming force. -- Two JSH quotes That merely shows that each real number can be approximated arbitrarily closely by rationals. It doesn't show that each real number is rational. Putting it another way, the property you described is equivalent to saying that the rationals are dense in the reals. It doesn't say that the set of rationals is equal to the set of reals. In fact, the rationals are countable, but the reals are uncountable. sqrt(2) is irrational. There is no surjection from the rationals onto the reals. Cantor's diagonal proof is one way to establish this. The transcendental numbers are not enumerable (or countable) at all. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. You have raised in interesting question. Let S be an ordered subset of the integeners. Let S||N be the set of elements of S <= N. We can define a density function as #(S||N)/N where # is just finite cardinality (a non-controversial quantity). Define D(S,N) = #(S||N)/N. Notice the D(S,N) is monotone non-decreasing with N. Now for different S we can look at the rate of growth wrt N. So it would make sense to say that for S1 and S2 if D(S1,N/D(S2,N) tends to 0 with N that S1 has a lesser density than S2 and we can order by density. We already have such a function for the density of primes wrt to the integers. Is there a general theory of such density functions? Comparison of densities would make some kind of sense. One might be tempted to look at derivatives of D(S,N) but that would mean fitting the discrete values of D(S,N) with a differentiable function. I do not see a unique natural way of doing that. May be you have a thought on that matter Bob Kolker Densities are a useful concept, but not one that generalizes very well to sets that are uncountable, or that are dense in an interval. As a general purpose measure of the size of sets, the concept has definite shortcomings. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. The only S for which D(S,N) is monotone non-decreasing is S = everything. I didn't write that. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. True, sorry about that. D(S1,n)/D(S2,n) = 0 you could say the density of S1 is less than the density of S2. I raised the question of whether there or other meeasures of how many or how much other than the usual measure, cardinality or volume measure (Lesbesque, Borel etc). Bob Kolker Allan C Cybulskie said: the same way on this. If we're crazy, at least we're not alone. It's nice to see someone else who trusts their intuition enough to question a formal system that violates it. :) -- Smiles, Tony than I is number. theings to system That used to be called critical thinking. I shudder at what has happened to that notion in this thread ... Generally, I just want them to give me a decent reason to accept the counter-intuitive results ... but that does not seem to be forthcoming. Allan C Cybulskie said: I think we have to be satisfied with the notion that cardinality measures don't distinguish between certain types of infinities that we would like to distinguish. I think mathematicians here have admitted that there is a difference between cardinality and other set measures, and that the conclusions of cardinality are consistent internally and not necessarily with these other measures. The real problem is mistaken conflation, as you say, of cardinality and the size of a set. I think everyone agrees that's just colloquial, and that cardinality is not THE proven correct definition of size. Does anyone disagree with that? -- Smiles, Tony If you include under the rubric size measure as a generalization of length, area, volume ..(onward to higher dimensions).. then what you say is quite true. There are different terms roughly related to size and one must carefully indicate what meaning is intended for a given context. Bob Kolker robert j. kolker said: -- Smiles, Tony Discussion, linux) When was trusting intuitions known as critical thinking? I shudder to think where you got your misplaced notions of philosophy. -- 'Every man who has ever lived holds tight to the belief that for him you will marry Guinevere. You do not want advice --- only agreement.' Merlin sighed... -- John Steinbeck Exactly what result do you consider to be counter-intuitive? That it's possible for a bijection to exist between a set and a proper subset of itself? That bijections can be used to establish an equivalence relation (a relation that is reflexive, symmetric, and transitive) between sets? That this equivalence relation is given a name? What? I think the only problem is that you are attempting to read some deeper meaning into cardinality that was never intended to be there and was never claimed. By the way, the distinction between countable and uncountable sets is of extreme importance throughout analysis. That is, it is vital to distinguish between the smallest infinite cardinality and all the others. I don't think it matters whether you accept the difference or not, but to those who work in the field, there are vital facts such as: Every countable subset of R has Lebesgue measure 0. No complete metric space can be represented as a countable union of nowhere-dense sets. Every uncountable sum of positive numbers diverges. Every uncountable subset of R^n has a condensation point. A Hamel basis for the reals over the rationals is uncountable. If a topological space does not have a countable neighborhood base at a point, then sequences are inadequate for studying convergence. We must use nets or filters instead. In short, it's a distinction that is omnipresent. That's why it has a name. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Dave Seaman said: That [0,1] has as many elements as [0,2], or that there are as many rationals, or evens, as integers. These conclusions seem to be contradicted by other ways of viewing the sets, externally to cardinality. If one wants to say that cardinality simply does not distinguish among these infinite sets, then I have no problem, but if one wants to claim that cardinality proves that these sets are equal in size, then I consider that an unwarranted conflation of cardinality with the general notion of size. I thought I had made that clear. It seems like it has been claimed, but that perhaps that claim is now being qualified or dropped, and cardinality admitted to be one of several measures of set size that do not all agree in their classifications. I am satisfied with that. Yes, I can see that countability is an important criterion for classifying sets, but it only has two posibilities that are mentioned, countable or not, so it's not a fine distinction, but a broad one. That's okay, but not fantastic. -- Smiles, Tony Do you consider it counterintutive that [0,1] has the same cardinality as [0,2], or that the rationals have the same cardinality as the integers? If so, why? Who says all views of a set have to give the same results? Doesn't that destroy the point of looking at sets in different ways? If you can find a single way to look at sets that makes such concepts as cardinality, measure, and natural density irrelevant, then you can publish and become famous. Good luck. I don't care what you think about the size of the sets. You have not demonstrated any problem in these sets having the same cardinality. If size makes you uncomfortable, then simply forget that anyone mentioned size and concentrate on cardinality instead. I didn't say those were the only two cardinalities. I said that the distinction between countable and uncountable sets is an important special case. But finite sets also have cardinalities, and it is likewise possible to distinguish between different uncountable cardinalities such as aleph_1 and c. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Dave Seaman said: No I find it perfectly intuitive that a method which glosses over the details of its own method fails to make obvious distinctions. However, [0,1] is half as bigulous as [0,2], despite being equicardinal. They are, beyond the shadow of a doubt, equicardinal, but not equibigulous. Or, forget that anyone mentioned cardinality, and concentrate on bigulosity. Right. I just said countability distinguishes two classes. Cardinality Distinguishes a hierarchy within the uncountables, but nothing further within the countable infinities, and no ratios between set sizes that aren't infinite. -- Smiles, Tony Discussion, linux) What does glosses over the details of its own method mean? The fact that you have a new word doesn't remove the need to define it. -- Jesse F. Hughes Ultimately, I can bring the entire mathematical establishment to its knees... Live in a fantasy world if you wish, but to me that's just an expression of your intellectual inferiority. --James Harris Jesse F. Hughes said: It means it uses reorderings and ignores the effect of those reorderings on the bijections, as I've said several times already. You never commented on the definition I posted. Why is that? -- Smiles, Tony Discussion, linux) Sorry, which definition is that? If I failed to comment on it previously, then I promise to comment on it now. Just give me a pointer to it. -- There was an accident in the air. There was a sign saying, 'Planes don't go here. The clouds have to be fixed.' -- Quincy P. Hughes applies the lessons of Dutch train travel to pretending about airplanes. A set glosses over the details of its own method when it doesn't define its own method in universal terms. Discussion, linux) Have I missed something? -- Jesse F. Hughes This Trojan appears to utilize a function of the Windows Media DRM designed to enable license delivery scenarios as part of a social engineering attack. -- MS candidly explains the role of DRM licenses It's possible unless we assume CH. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Discussion, linux) You and I must parse the assertion differently. I wouldn't say it is possible to distinguish them unless they are indeed different. But they're not different unless we assume NOT CH. For me: it is possible they are different is not the same assertion that it is possible to distinguish them. But this is natural language semantics and not math, I guess. I sure didn't read it your way until your followup. -- Yup, as far as I'm concerned, if you live out your lives smiling the entire time full of pride in your *believed* accomplishments, when you never had any, well that's ok with me. --James Harris, a man of remarkable accomplishments. I consider that they have different definitions and therefore are conceptually different as long as we take no position on CH. I can certainly distinguish between the definitions. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Discussion, linux) Jeez, but are you this utterly ignorant of everything that you criticize? Know what? There is more to cardinality than countable/uncountable. After Aleph-null (the only countable cardinal) and Aleph-one (the first countable cardinal) there are lots of other cardinals. Dozens even. Maybe more. Dave mentioned one important distinction (countable vs. uncountable), but he did not say or imply this was the end of cardinality. Stop giving philosophy such a bad name. Go learn something before you pass on stupid pronouncements about how you know more about mathematics than mathematicians. Really. Please. -- Jesse F. Hughes Really, I'm not out to destroy Microsoft. That will just be a completely unintentional side effect. -- Linus Torvalds Jesse F. Hughes said: Shouldn't that read Aleph-one (the first NONcountable cardinal)? What is there between Aleph_0 and Aleph_1? Aleph_0.5? No, cardinality specifies a hierarchy of uncountable infinities. Does it distinguish between countable ones? Pretty please, with sugar on top? And a cherry? Come on, ask nice now..... -- Smiles, Tony Exactly what results do modern mathematikers offer as counter intuitive? They seem to be the experts on the subject. Not so much that as the terminological regression of otherwise universal concepts such as cardinality. Actually the meaning of cardinality is quite clear and simple and was always intended to be universal and not at all what modern mathematikers make of it. The only counterintuitive result in modern mathematics is that modern mathematikers subvert universal meanings and then claim they're making things clearer. It's easier to work with tangents. A fascinating example in aid of nothing to justify parochial definitions of cardinality. I think the Banach-Tarski paradox counts as counter intuitive. Cardinality was not a universal concept until Cantor defined it. Can you show me a discussion of cardinality of infinite sets that precedes Cantor. It is quite clear that you are not at all interested in discussing the concepts but simply in being argumentative. This is my last response to you. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. And I thank you for your thoughts. Now perhaps if you could just define counterintuitive for us in universal terms, perhaps we could all come to some conclusion regarding modernmathspeak. Oh, horse. Cardinality had no definition until Cantro defined it? I can't show you a discussion of sets period prior to Cantor. Can you show me how the answer to this question resolves the universality of sets and the cardinality of sets exactly? Good. Now we can on to resolving issues of science in universal terms. Yes, it was. Cantor re-defined it for mathematicians who wanted to do absurd things with infinities. I take your word that Cantor was responsible that re-definition of cardinality. Concepts can be discussed on other terms than the narrow views you hold. -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy Albert Wagner said: You know Albert, that quote of yours keeps reminding me of our weeks-long argument...... ;) -- Smiles, Tony What to tangents have to do with arbitrary sets? What could tangent possibly mean in that context? Bob Kolker If you can explain what mathematics has to with aribitrary sets, perhaps we could get to the bottom of your little puzzle. If the cardinality of arbitrary sets refers to the count of elements in the set, I daresay, there is a tangent to the count but possibly not to the count of tangential mathematikers in the arbitrary set. That is meaningless drivel. Bob Kolker Lester Zick said: The tangent doesn't help distinguish between arbitrary sets. it distinguishes between subsets of a given arbitrary set, okay? Sheesh! -- Smiles, Tony Discussion, linux) Fine. Let's take the given set to be powerset of N. Consider the set { S | N S is infinite } (the set of all subsets of N such that their complements are infinite) { S | S c Prime } (the set of all subsets of the set prime numbers) Help me out here. What are the tangents to these sets and which one is bigger? -- [Criticizing JSH's mathematics will result in] one of the worst debacles in the history of the world. It is foretold in most mythologies and religions. And yes, you are the ones, the cursed ones, who destroy the world. --James S. Harris reads from the Aztec Book of the Damned Mathematicians Jesse F. Hughes said: Do these look like mapping functions between numeric set members to you? Please give me the derivative of an octahedron. The set of all subsets is a powerset, so you're trying to partition the powerset into finite and infinite members. I do not see a differentiable function there, do you? -- Smiles, Tony Discussion, linux) You said: it distinguishes between subsets of a given arbitrary set. Powerset of N is a given arbitrary set. What is the problem? -- But you people are scum of the earth who pretend to be something that is clearly beyond you--real mathematicians. I wouldn't be having these problems with Gauss or Euler. I wouldn't be having these problems with Fermat or Archimedes. -- James S. Harris on pretending And on yours for the counting of universal ideas. The issue of count-intuitivity is vexing. Why should the power of a mathematical concept or system by limited by intuition when logic can carry out the consequences of the axioms of a system? It is true that intuition can sometimes help find a proof or solve a problem. but the limits of intuition can equally inhibit understanding of the consequences and blind the intuitive to other non-intuitive possibilities. Topoloy is full of example of constructs and structures that boggle the visual intuition. The solution is simple. Do not rely totally on visualization. Use analogy or metaphor instead. It seems to me that progress in mathematics and physics has been made largely by abandoning intuition or common sense when it becomes an impediment. Bob Kolker robert j. kolker said: You know, it's really not intuition that I am speaking of. I used the word counterintuitive with my tongue in cheek, because that's the way such conclusions are characterized by those drawing them. The problem is really one of consistency between methods of drawing conclusion regarding infinity, and between the implications of those conlusions and what we actually observe. Intuition is a good place to start when evaluating a situation, but then of course one needs to check their intuition rationally to see where it really leads. This interaction between wholistic associative thinking and rational thinking is vital to creativity, I believe. -- Smiles, Tony Oh, yeah, that's rich, Bob. You got that right, when it becomes an impediment to science. Daryl McCullough said: I don't have a problem when cardinality says more elements. I have a problem when it says same number of elements, when it means same cardinality measure according to what we have defined, whether that actually reflects any particularly exact measure of the size of the set or not. If that's what you mean, by all means use the word cardinality as defined, instead of asserting that cardinality measures prove that there are as many integers as rationals. It doesn't. That's a non-trivial point. -- Smiles, Tony Well, I tried to give this thread a more appropriate title, since we have long left the subject stated, but that post few responses. An open letter. Tony- I do hope you read my previous response to Allan C. C., which I redubbed with the subject 'Number of elements'; was: Distinct linear orderings on Z. I made many remarks and questions there which I hope you will consider and answer in these newsgroups. I will be repeating some of these here. I have just reviewed all your posts in this thread. (I have not been reading the Epistemology 201 thread at all, if that has anything to do with this one.) In this post, I try to respond to your remarks both generally and specifically. No one disagrees about what the number of item in a finite collection means. The problem you address here is how to discuss the number of items in an infinite collection. Once you have even started talking about infinite collections, you have entered the realm of mathematics, and definitions must be precise and rigorous in the sense thereof. Even if informally, modern mathematics considers number of elements to mean cardinality. You disagree with this identification, for you find some of its implications counterintuitive. It behooves *you* to provide a precise definition of what you mean by number of elements or by collection a A has more elements than collection B, to show that this definition is indeed well-defined, and to justify it. Such a definition should be able to compare any two collections, finite or infinite. You have yet to explicitly specify what your definition is, and to show that this definition is free of counterintuitive implications. There are two general weaknesses in the arguments you have presented thus far. 1) You assume that your intuition, derived from the observation of finite collections (which are observable in the physical world) must extend to infinite collections. E.g., if A is a proper subset of a finite set B, then B has more elements than A. Ergo, you say, the same implication must hold for infinite B. 2) You confuse different ways of assessing the *size* of collections. I address these two items individually. 1) I don't know if we can reach common ground here. I can only offer the following examples, generally accepted by well-trained mathematicians today; many of these have spent time considering the matter. a) The infinite bus. As does the next, this item addresses the comparison of N = {0, 1, 2, 3,...} and N {0} = {1, 2, 3, 4,...}. SJH: ACC: SJH: TO: - You are correct in saying that your first sentence is a non-answer. For example, if the passengers get on sequentially, and passenger n+1 finds his seat (n+1)^-2 time-intis after the nth passenger, then it takes finite time to load the bus. - The conclusion is not absurd to me (and to many others). - The real answer is: The number of seats has not changed at all; why has the number of passengers? - To answer your last question, why not? I.e., why should infinite busses be the same as finite ones in all respects? b) You claim than N = {0, 1, 2,...} clearly has more elements than N {0} = {1, 2, 3, ...}. We will discuss below the concept of containment qua comparison of size. For now, I offer you the question I posed in an earlier post. Compare the sizes of the two sets in each comparison. c) I think you agree with Allan that N = {0, 1, 2,...} has half as many elements as {0, 2, 4, 6, ...} = {2n: n in N} (often denoted 2N). Do {(1,0), (1,1), (1,2),...} = {(1,n): n in N}= {1} x N have the same number of elements as {(2,0), (2,1), (2,2),...} = {(2,n): n in N} = {2} x N? I don't think anyone can justify a negative answer. Yet replace each ordered pair by the product of its components, and, according to Allan and you, you have two sets which are no longer the same size. Is this observation not counterintuitive to you? d) Do you agree that if we list N as {0, 1, 2, 4, 3, 6, 8, 10, 5, 12, 14, 16, 18, 7...} (so that we increment the number of even numbers before each odd one), it does not change the number of natural numbers from that when we list them canonically {0, 1, 2, 3, 4, 5,...}? If so, why do you say listing the nonnegative rationals as {1, 2, 1/2, 1/3, 3/2, 3, 4, 5/2, 2/3, 1/4, 1/5,...} (the zig-zag order) changes their number? Here is a characteristic *common* with finite collections: Changing the order in which you count the items does not change the number. e) Even without delving into the proof of the Cantor-Schroeder-Bernstein Theorem, we can make the following argument. Since the set N+ of positive integers is a subset of the set Q+ of positive rationals, the number of positive integers is less than or equal than the number of positive rationals. Given a positive rational number r = p/q in lowest terms (i.e., positive integers p and q have no common factors other than 1), pair r with the positive integer 2^p * 3^q. We thus show that the number of positive rationals is less than or equal to the number of positive integers. Thus, since each of these numbers is less than or equal to the other, the two numbers are equal. After all, the same is true of finite collections. f) Consider the two sets A = {1, 2, 3} and B = {2, 4, 6}. Each of these have three elements, hence these two sets have the same number of elements. I.e., if A is a finite set of integers (or rational, real, or complex numbers, for that matter), then A and B = {2x: x in A} have the same number of elements. Why should this property change when A is an infinite set, e.g., N or [0,1]? Considering your earlier objections and items (d), (e), and (f) (and even (b) and (c)) above, I ask you: Which properties of finite sets should generalize to infinite sets, and which should not? 2) Though you may not recognize these identifications, you have presented several different methods of assessing the size or relative size of collections: a) set containment, b) count of elements, c) length of intervals, d) natural density, and e) topological density. While there are some correspondences amongst these in specific cases, the correspondences do *not* generalize universally. Which is the appropriate generalization of the concept of number of elements? Well, I need to get to bed, so I will continue this later (or let others to do it for me), maybe. (Some topics: counting and length are two special cases of measure; not all subsets of R have a natural density, and there exist nonempy sets with natural density 0; natural density does not allow the comparion of disparate sets, e.g., P(N) vs. R; the Cantor set is nowhere dense in [0,1], yet by consideration of ternary and binary representations, there are just as many elements therein as in [0,1] certainly more elements than the rationals; the dyadic rationals are also dense in R.) Still, I think you have enough to chew on and respond to in this incomplete post. I do add some meta-remarks and one not so meta: 1) I have the impression that Tony and Allan CC have genuine argiments to make and are sincere in engaging in discussion and willing to consider alternative positions. It is for this reason, and the fact that others may wish to consider this discussion, that I participate here. 2) I find it remarkable that Tony and Allan insist that some infinite sets have more elements than other infinite sets (in cases where most mathematicians disagree). More often, people object to the concept of different sizes of infinity (e.g, that there as many rationals as there are integers but there are more reals than there are rationals). 3) Virtually all mathematicians today agree that bijections and cardinal numbers provide the appropriate method to generalize the concept of number of elements. While it is extremely commendable that you should consider this question on your own, don't you think it is a bit outrageous of you to insist that mathematicians who have spent their careers thinking about such things are wrong? Might a better attitude be to investigate how such mathematicians came to such conclusions? 4) In case Tony and others missed it, here is an argument (from an earlier post of mine) to justfy the use of bijections and cardinality to generalize concept of number of elements: Suppose you are in a store buying some items and your total comes to $6.72. (Pardon my US currency bias.) You happen to have lots of singles (= one-dollar bills) in your pocket, so you decide to pay with them. What do you do? You count them out - 1, 2, 3, 4, 5, 6, 7 - and hand them to the cashier. I .e., you have set up a bijection between the set {1, 2, 3, 4, 5, 6, 7} and the collection of bills you hand to the cashier. That is how one counts, i.e., determines the number of items in a collection. For some reason he did not make clear, Allan does not accept this description. We say two finite sets have the same number of elements if they each can be put in bijection with the same natural number. (Formally, finite n is defined as the set {0, 1, 2, 3, ..., n-1}.) This implies there is a bijection between the finite sets. E.g., count off the numbers in the set A = {17, 212, 402, 5135, 222001, 333333333, 1200000001.} This sets up a bijection between A and 7 = {0, 1, 2, 3, 4, 5, 6} (call it {1, 2, 3, 4, 5, 6, 7} if you want), and we say that A has 7 elements. Formally, we can construct prototypical transfinite numbers, e.g., infinite cardinal numbers. One example is aleph0 = {0, 1, 2, 3, ...} (also called omega or, in different contexts, N). (The natural numbers are the finite cardinal numbers.) There exists no bijection from one cardinal number to a different cardinal number. Since each of N = {0, 1, 2,...} and N {0} = {1, 2, 3,...} can be put in bijection with aleph0, we say the number of elements in each set is aleph0. That is, we can conceptually count off the elements in each set by going through the elements of aleph0 in order. (Just like we determine that {3, 5, 7} and {2/3, 17/19, -2} each have 3 elements by counting off 1, 2, 3.) In particular, each of these sets have the same number of elements, and they can be put in bijection with each other. Let A and B be sets with B infinite. If there exists a bijection from A to B, then there also exists a nonsurjective injection from A to B as well. (More precisely, you need B Dedekind infinite; the two concepts are equivalent in the presence of choice.) Such a situation cannot happen when B is finite. E.g., consider two maps f and g from N {0} to N: f (x) = x, the inclusion map, and g(x) = x-1. f is an injection but not a surjection, while g is a bijection. Why does the existence of f lead you to say that N has more elements than N {0}? Because that is true for finite sets? You seem to insist that this distinction between finite and inifinte sets renders bijections inappropriate for counting. Put another way, There exists a nonsurjective injection from A to B is not the negation of There exists a bijection from A to B unless you add the condition that B is finite to both. (Alternatively, you could add the condition that A is finite.) What makes you prefer the former statement as a basis for comparing the number of elements in a set? -- Stephen J. Herschkorn sjherschko@netscape.net Stephen J. Herschkorn said: Bigulosity Measure of Infinite Sets: Two sets have the same bigulosity, or are equibigulous, if there is a direct mapping between successive elements of the sets without any reordering of the sets, that is, the mapping function f(x)=x+c, where c is constant. The relative nth partial bigulosity of two sets is defined to be the pair (n,nth derivative of f}, and is valid for n=0 to x, where the xth derivative of f is constant. This final constant nth derivative, along with n, denote the topmost measure of relative bigulosity for the sets, which in combination with all previous pairs of derivative levels and derivatives, constitute the full expression of relative bigulosity of the two sets. The bigulosity of finite sets is defined the same as cardinality: absolute number of elements in the set. The reals in [0,1] and [0,2] are not finite sets. Nor are the integers or rationals, or the evens. Come again? The inconsistencies I have mentioned all concern infinite sets which can be dealt with in other ways, which don't agree. It's not a matter of intuition, so much as consistency. So the passengers get infinitely fast as they go to infinity? Is this a relativistic bus? It was a non-answer because it assumes some actual time in the process. It was a joke. Like that tree in the woods, that falls on a pile of marshmallows, in front of a deaf guy and a dog that hears, but happens to be sleeping....... Oh, well, if everybody else says so........ Because one got on the bus, and none got off, which means an additional passenger is on the bus, because one was added, and none was subtracted. But you did that in second grade..... That is not a valid question. If you want to make rules, and then make exceptions to them, either you have to generalize the rules, justify the exceptions, drop them, or admit that you're creating an arbitrarily inconsistent system. I haven't seen a good justification for why adding something doesn't somehow make it more, except oo+1=oo by definition, which is just another arbitrary definition. You can certainly draw a bijection between the two. Of course you have to realize that your mapping functions f(x)=(x,x+1) and g(x)=(x-1,x) do not map a number to a number, but one number to two numbers, and are therefore two different mappings. When you look at the original two sets, without the artificial superposition of them, the first includes all that the second does, plus the additional element of 0. You add one more element, the set is one element larger. By Bigulosity Theory, since the mappign function is f(x)=x+1, the topmost relative bigulosity is (1,1), so their overall relative bigulosity is 1, and yet the term (0,1) is a significant offset indicating a finite difference between the infinite sets. I think I am making some small error in my calculations here, as I invent them, but I think Bigulosity is a valid concept. No, the evens have half as many as the naturals. You got it backwards. If half of the naturals are even, then aren't the evens half of the naturals? Can you really argue about that? Isn't it true by definition? Where, besides infinite cardinalities, have you ever seen this simple form of logic violated? Changing the order to derive a mapping function changes the measure of the realtionship between the sets. I would agree with that straightforward argument. Then py=qx, and 2^p*3^q = 2^(qx/y)*3^q = 2^(x/y)^q*3^q = (3*2^(x/y))^q Is that equal to 2^x*3^y? There is a reordering the likes of which I have been warning against, so the example doesn't really work for me, and is inconsistent with Bigulosity Theory. If you want to describe a set of the evens over a domain twice as big as a domain containing a set of naturals as having the same bigulosity, I have no Adding means making more. Taking away means making less. Proper subsets have fewer elements, and proper supersets have more. It is just this universal generalization that I regard as necessary to avoid the acceptance of absurd conclusions. Bigulosity's a work in progress. After all, the term was only coined today!! Whaddya want??? :) during the day (while wotking) so until I get my 'puter together with email at home, I can't respond at night. Sorry to any who have wondered why I don't say anything all night. And I thank you. They call me the backwards salmon. LOL. Yes, everyone, like my mother for instance, say, but isn't infinity just infinity, and that's that?. I learned about Cantor a long time ago, and found the ideas very exciting, but always felt that some of the conclusions were wrong. I have worked on infinities and zeroes from other perspectives that seemed more illuminating, and I guess when the topic was brought up, it reminded me of my objections, so what the heck, let's follow that tangent if it makes for good brainercise. I've learned a lot in the process, but nothing has managed to change my mind. It's really just forced me to try to formalize a better system. I learned about it in high school. I understand that it was a big step in the right direction, but I can't help but seek better consistency than I see in the menage of methods that reach different conclusions. I don't think that I can abandon conclusions based on simple logic for ones based on an unproven complex method when they disagree, without some explanation as to why simple logic should fail. For lack of such explanations for the exceptions to rules, I can only view them as an indication of contradiction. Of course, if you have a five and two singles, that's still a set of seven dollars, isn't it? In this case, would you consider the set of seven singles to be equal to a set of seven five dollar bills? Tell you what, you shop at my store and I'll give you change in pennies!!! ;) I think you have just pointed out the limitation of cardinality. It certainly cannot distinguish between two sets that are not in an infinite ratio, at least, with respect to size. If it is possible to have an infinite set twice as large as another, the bijection method eliminates that factor of two, so cardinality loses that piece of information. To claim the information never existed is unfounded, using a method not designed to detect it. If I look at the sky with binoculars, can I insist that there is no difference between stars and galaxies, or that galaxies don't exist? Perhaps I'd need a better tool. Stephen, I do thank you for your input, and hope my comments are clarifying, even if they're not decisively airtight and conclusively bullet-proof. Whew, back to work (boohoo). -- Smiles, Tony Mathematikers use parochial definitions then pretend they've said something of universal significance. Stephen J. Herschkorn said: Excuse me but I never said they were uncountable. I said that the set of rationals, by logic separate from countability or bijections or any of those methods, is clearly larger than the set of integers, and that the conclusion that it is not derives from a contradiction inherent in the reordering of numerical elements that derive their meanings, which are being used for the mapping functions, from their original natural ordering. Consistency is a good thing. The more exceptions one has, the shakier the system. Some basic set relations should have priority over more specialized complex approaches. Kind of an Occam's Razor kinda thing. That is the case for finite sets, Why should it not be true of infinite sets, bijection aside? I am working on the alternative, and this discussion is helpful. The correspondence method seems to be the modern method of choice. Does that mean it's the last word on the subject? I hope not. If the reals are distributed consistently throughout the number system, then [0,1] should have half the number of elements as [2,4], as reflected by the proportional lengths of the sections of the number line they occupy. No, since that process would take an infinite amount of time. Okay, that's a non-answer, but I have seen this example before. One version of it left an infinite pile of children's fingers at infinity to draw a 1-1 correspondence between the children and their fingers. It's kind of absurd, don't you think? The real answer is, there is one more passenger on the bus, if one got on and none got off, just like for finite buses. Why should this be different for infinite buses? -- Smiles, Tony Dave Seaman said: Ah, but it is you who misunderstand the argument, for the ordering does make a difference which cardinality ignores, as you have just said. This is what I believe causes cardinality to be interpreted to have counterintuitive conclusions that beg correction. -- Smiles, Tony Discussion, linux) Of course cardinality ignores order, because it is a measure of size of arbitrary sets. If you want to take order into account, that's fine, but the result isn't a good approximation of set size. -- It seems to me that in wartime Americans shouldn't be attacking each other in this way on a *worldwide* forum. Then again, I know I'm an American, but I have no way of knowing that you are, which would explain a lot. --James Harris, on why Yanks should accept his proof number of elements to mean something different from cardinality. Since then I have asked repeatedly for a definition of number of elements, but none has been forthcoming. There is a different kind of mapping called an order isomorphism that applies to ordered sets. Such a mapping is a bijection with the additional property that it preserves order. Therefore, you have a choice: you can talk about plain bijections, which determine cardinality, or you can talk about order-preserving bijections, which determine order type. By convention, mathematicians understand the phrase number of elements to refer to the former, not the latter. If you want a different meaning, it's best to use the technical term and talk about order types, not the misleading term number of elements. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Dave Seaman said: basic definition, but I wonder if this approach has been applied to deriving finite ratios between the sizes of infinite sets, such as between [0,1] and I read about order types, this extension has not been realized. Is there another term I should be looking for? It seems to me that the term number of elements, being constructed from the fairly simple concepts of number and thing, is a rather general term. The more exact term as it stands is cardinality, so it seems to me that if that is what one is referring to, then that's the word they should use. To equate cardinality with number of elements for infinite sets, to the exclusion of any other definitions or approaches to defining that vague term, is a mistake as I see it. It's not that cardinality is a mistake, just that assumed implication, and I am not at all convinced that it's a semantic or philosophical matter, when talking about consistent mathematical results. -- Smiles, Tony There are other names for these concepts. There is a branch of analysis called measure theory that considers a generalization of the concepts of length, area, and volume. If we use m to denote Lebesgue measure, then here are some facts about the measure of various sets of real numbers: (1) m([a,b]) = b - a whenever a < b (the measure of an interval is its length), (2) m(A) = 0 for every countable set A (such as the set of rationals, the set of integers, or the set of primes), (3) m(C) = 0, where C = the Cantor set (an uncountable set). (4) there are sets that are order-isomorphic to the Cantor set but have positive measure. Another term that is related to things you have said is density. For example, you used density when comparing the sets (a) multiples of 1000, (b) squares, and (c) primes. All of these sets have measure 0, because they are all countable, but they have different asymptotic densities. This is a concept that depends on the ordering of the real numbers. So the basic problem is that there may be more than one way to compare sets. Mathematicians avoid confusion by assigning different names to these different concepts. It's possible that sets such as the reals in [1,2] and the rationals in [2,4] may give different results when compared in different ways. But you are overlooking the fact that number of elements has no mathematical definition and no one is actually claiming to have proved anything about the meaning of number of elements as a phrase in the English language when applied to infinite sets. You are confusing formal mathematical terminology with informal usage. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Not true. I mentioned specificall that cardinality and number of elements mean exactly the same. If you choose to ignore what you read very few definitions tend to be forthcoming. I was talking about my response to Allan Cybulskie, in which I incorrectly assumed that he was using number of elements to mean cardinality. He wasn't. If you agree with me that number of elements means the same as cardinality, then my request for a definition does not apply to you. I specifically made that request only of those who claim that the meanings are different. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. I apologize. I do define the two the same. But there still remains the issue of which sameness is meant. I mean that cardinality means the number of elements and not matching bijections which I consider to be a different concept altogether more akin to L'Hospital's Rule in ordinary mathematics. No apology is needed, because I have been guilty of confusing identities and positions of various people in this thread, myself. But one point of clarification. Cardinality as a mathematical term means what I said it means. It does indeed depend on bijections. If you claim number of elements (which is not a mathematical term) does not have anything to do with bijections, then you are claiming that the meanings are different. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. You might be interested in a collateral reply to Tony on this subject. Cardinality in universal terms means exactly the same as number of elements. When cardinality is applied to undefined numbers of elements the modern mathematical meaning of bijective mapping simply refers to the slope or tangent of the set instead of the set itself, which is a restricted parochial definition and not universal. In other words, the modern math approach to set analysis in terms of cardinality is done in non universal terms whereas the conventional interpretation of cardinality in ordinary mathematics is universal in nature. Basically what I'm saying is that cardinality and number of elements are identical concepts in universal terms of ordinary mathematics and only become different when cardinality is interpreted in terms of slopes or tangents as it is in bijective matching in set theory in modern math. And of course it is this bifurcation between universal and parochial applications of the single term, cardinality, that leads directly to the nominally counterintuitive results in modern math. A definition that works only for finite sets is not universal. Cardinality applies to arbitrary sets. How do you compute the tangent that recently came up in sci.math? Or how about the tangent of the set of all subsets of R? I have quoted you the definition of cardinality, and it says nothing at all about tangents. As best I can understand it, you are using universal to mean applies only to finite sets, and parochial to mean applies to all sets, whether finite or infinite. Huh? -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. And a definition that applies to all things is universal. And what, exactly, is an arbitrary set? How about the derivative? Not quite. As I plainly state gauging infinities one to another by means of tangents is a well established principle. Modern math is not nearly so useful as ordinary math but considerably more complicated. I'm not sure I know any definitions that apply to all things, but I know some that apply to all sets. A set that is not presumed to be a subset of any particular set, such as the reals. There are lots of things in mathematics that are not numbers. For example, the set of all bounded linear operators on a Hilbert space, or the homology group of a torus. Cardinality has nothing to do with derivatives. Tangents have nothing to do with cardinalities, and they don't even exist if the sets you are discussing are not sets of numbers. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Well, see, here the problem is that mathematikers draw definitions in parochial rather than universal terms.When asked to define cardinality mathematikers begin the definition sets have cardinality when . . . which presumes all kinds of nonsense including that what one is analyzing are sets to begin with. If mathematikers want to analyze sets, that is certainly their business. But that doesn't mean that definitions drawn in parochial terms are appropriate definitions in general universal terms. I don't know if mathematikers remember their grade school teachers' admonitions regarding definitions of the form: A is when . . . The appropriate way to define cardinality universally begins cardinality means . . . or cardinality is . .followed by a series of predicates in order of descending universal significance. Funny, I could have sworn the parochial definition in set theory talks about matching bijections. Matching bijections don't have cardinality if there is no continuity between elements matched. If there is a matching bijection there is continuity in the matching bijection and a tangent to the set. Lester Zick said: I must concur with Lester on this one. There is sense here that's important, as I've expressed several times, with the notion of derviatives, and probably L'Hospital's Rule. Lester has a point, which may be hard to see, especially when he keeps sticking it in your eye! ;) -- Smiles, Tony I have presented no such definition. The sloppy language is entirely your own. The definition that I gave you on more than one occasion is: Definition. The cardinality of a set X is the least ordinal that can be mapped bijectively to X. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Yeah, Dave, let me explain something to you. Contrary to what you state, the sloppy language is entirely that of mathematikers, not me. You indicated in your prior reply that you would not be replying further to my posts and yet you persist. Allow me to clarify my position with respect to condescending assholes who consider the universal definition of cardinality didn't occur until Cantor came along that you have no idea what universal knowledge is or how to define it and that I do. Then allow me to cordially invite you to go yourself. Discussion, linux) Wow. I thought that Tony, a relatively sober guy[1], was kidding when he said something about the number of elements of N as the derivative of the number of elements of powerset of N. I have to revise my opinion. Maybe he was serious and seriously influenced by Zick. And maybe he thinks that a constant like P(N) should have non-zero derivative and also that the derivative of 2^x is x. This is not respectable philosophy. This is pathetic. There are philosophers of mathematics. These guys bear no particular resemblance to a proper philosopher of mathematics[2]. Footnotes: [1] It's easy to appear relatively sober when standing next to Lester Zick. [2] Me either. I'm interested in the topic, but I have never contributed to it. -- This sucks, said a Pennsylvania State University student [...] Why can't the college let me do what I want to do with my computer? The school computer security guys are being way more annoying than the spyware was. -- A student pines for his disabled spyware Jesse F. Hughes said: Eggsqueeze me, but I never said the things you just stuffed into my mouth. I am not led by Lester. I just understand some of what he is talking about, and agree with some of that. In other areas we disagree, like dimensionality, and the virtue of empirical evidence and the bottom-up approach. I think that Lester's vision of infinities as infinitesimals is a little less clear than it could be, but that the concept of using calculus to resolve infinities is far from outrageous; that's what it was invented for. Read my preliminary description of Bigulosity and comment on that. Reserve judgement on the derivative of the powerset comment for a little bit, after all the power set is not a number, and the size of the powerset is a compound infinity, akin to a multidimensional infinity that can be differentiated to one fewer diemsnions. I may very well come to the conclusion that that's not the right description, and yet for now, I think it's right. Maybe it's the second derivative over the domain of the powerset...... -- Smiles, Tony Discussion, linux) The main thing I stuffed into your mouth is that you said the number of elements in N is the derivative of powerset N. You're right. You didn't say that. You said either [1,2] or (0,3) has something to do with the derivative of powerset of N in the following exchange. ,---- | It is the derivative of the power set of N. `---- Note: When asking how [0,1] compares (in terms of size) to the powerset of N, your answer is not even linguistically coherent. Fine. You are your own, self-determined and incoherent deep thinker. -- I have to break the code of how [mere humans] work, and I have made a lot of progress over years of effort, and I feel like I am close to figuring out all the inner details of human wiring. -- James S. Harris on the extra problems of conveying his research Precise statements about aleph1 and c. aleph1 = c is an immediate theorem of ZFC + CH aleph1 <= c is an easy theorem of ZFC aleph1 < c is an immediate theorem of ZFC + not CH. aleph1 < c is consistent with ZFC aleph1 and c are incomparable is consistent with ZF + CH. (We discussed this a while ago. By CH, I mean that any uncountable subset of R has cardinality c.) aleph1 and c are incomparable is consistent with ZF + not CH. aleph1 and c are incomparable is consistent with ZF. -- Stephen J. Herschkorn sjherschko@netscape.net [. . .] Lester tends to prefer deep thinkers who can't quite come to terms with universal truth. Jesse F. Hughes said: And you are a mathematician? You certainly display the pervasive lack of humor that seems to afflict that species. I think that's what makes a good -- Smiles, Tony Discussion, linux) Depends on who you ask, I suppose, but probably not. -- Run mathematicians, RUN!!! I'm coming for you. It may take a few months, but I'll get [computer verification of my proof] and then your lives will be ended as you previously knew it. -- JSH meets PVS You undoubtedly have many opinions worth revising. We are all pathetic indeed. Just not quite so pathetic as conventional mathematiker definitions of a cardinality which is certainly parochial enough that I expect the pathos extends to modern mathematics in general. Who undoubtedly awaits your particular anointing. It is? Please produce a reference in either a standard treatise or in the peer reviewed mathematical literature. Bob Kolker robert j. kolker said: L'Hospital's rule resolves ratios between infinities or zeroes to finite terms. -- Smiles, Tony I thought your comments were the literature, Bob. Certainly your definition of circles has lightened the burden of those who thought they were traveling on spheres in three dimensions instead of two. I am asking for a reference to the literature. Will you give it or not. Can you give it or not? Bob Kolker Of what possible use would /that/ be? -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy Dave Seaman said: I think what Lester is referring to is the idea that the rate of increase of the members in a set defined by a function is inversely related to the size of the infinity. For instance the evens are defined by the function f(x)=2*x on the integers, which has twice the slope of f(x)=x, and therefore repesents half as large an infinity. When the ratio is oo/oo or 0/0 for x=oo, one should be able to apply L'Hospital's rule to resolve the slope and calculate a tangent. I think parochial is Lester's poetic way of saying local, perhaps referring to local limits determining slopes! Or maybe not.... -- Smiles, Tony It's Lester's poetic way of saying moder math sucks non universally, Tony. Lester Zick said: So it only locally sucks, say, as x approaches cardinality? ;) -- Smiles, Tony That approach does not apply to all sets. It doesn't even apply to all subsets of the reals. How do you compare the size of the set of rational numbers with the set of algebraic numbers, for example? -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Dave Seaman said: That's true, it can't be applied to all sets. I doubt there is one approach that will work with all sets one could possibly define. Part of a cohesive system is a way of identiying which types of sets can be compared using which kinds of methods, and what types of results are possible. Still, I think gross approaximations at least can be made for almost all sets, and this approach applied in a large number of cases. Notice rationals are not defined by a function of reals to reals, are they? That does confound this approach, at first glance. thinkthinkthink........ -- Smiles, Tony Whoaaa!!!!!! What the hell about the approach accepted by the majority of mathematicians - i.e., the use of bjiections and cardinalities??? Or are you saying that there is no such method devoid of implications which *you* find counterintuitive and, you infer, objectionable to the point of rendering the method invalid? -- Stephen J. Herschkorn sjherschko@netscape.net Stephen J. Herschkorn said: Given the variety of sets, and sets of sets, and sets thereof, I am sure there are some that will defy many attempts to capture their essence. Is every function on x differentiable? No? Does that make differentiation wrong and less universal for functions on x? I never said cardinality is invalid. I said that claiming it is the sole measure of size for infinite sets is unfounded, and pointed to examples were it disagrees with what seems like more basic logic and more straightforward understanding. But I've only repeated that a dozen times, so I can understand the misunderstanding. -- Smiles, Tony It just applies to all continua. I'll leave the counting of hermit operators to those with fingers and toes. Lester Zick said: I may regret asking but....lawnmower accident? Piranha attack? -- Smiles, Tony