mm-1999 === Subject: Re: abundance of irrationals >> A short program of about 10 lines in Python computes Int(Pi*10^100) >> quickly & easily on my PC, returning >> 31415926535897932384626433832795028841971693993751 >> 058209749445923078164062862089986280348253421170679 > Actually, if I feed my program with the string > 104719755119659774615421446109316762806572313312503527365831486410260546876 20696662093449417807056893 > then it tells me that this number is an integer. I am assuming that > Int is supposed to be rounding down. >> You've lost me. The decimal string I posted is >> the first 101 digits of pi, i.e. floor(Pi*10^100). > His original question was whether floor(Pi*10^100)/3 is an integer. > The string I write is just that. Your string is 3 times that. I was replying to, and had quoted, this by ... The number 10^10^10^10 does exist. You can use it for calculating. Numbers like Int(Pi*10^100) will never be available. hence, they donn't exist. But you deleted that quote. Elsewhere in this thread (and about an hour before you, apparently), I posted Int(Pi*10^100)/3 = 7^2*13*1237469203* 13284792831169045259982534129276001843307875 327059949284940553259037316230601022146398363. and quoted the appropriate material I was responding to. --r.e.s. === Subject: Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false? <41eb3a33$0$563$b45e6eb0@senator-bedfellow.mit.edu> <41ee87ee$0$562$b45e6eb0@senator-bedfellow.mit.eduI didn't say that ZFC is consistent iff it has models. (Though that's >obviously true). What I did say is that ZFC is true relative to a >fixed structure S iff S is a model for ZFC. > I wasn't intending to argue that in fact you had made that statement, just > that you were happy making statements of that type. This was an attempt > to show you that you are guilty of exactly the same sin you're accusing > me of. In your parenthetical remark here you say, that's obviously > *true* (emphasis mine). What do you mean by true? We have the sentence > (*) ZFC is consistent iff ZFC has models. > You're comfortable asserting that (*) is *true*. I ask, true in what > model? Model of what system? of PA? of ZFC? If you wanted to be really clever, you could have asked why I think ZFC is consistent iff ZFC has models when the proof of that fact requires AC instead of this. ;-) Now that I know that, I'm not so sure that ZFC is consistent iff ZFC has models is true. However, let's assume that AC holds. (So we're working within ZFC) The class of models of ZFC, being non-empty, is a model for FOL + ZFC has models. The completeness theorem gives you ZFC is consistent as a consequence of FOL + ZFC has models. If you're asking me to exactly pin this down, it's going to be an ugly beast. But you can always create a model for a theorem by throwing the theorem's hypotheses in as axioms with FOL. >You complained about > the cartesian product of nonempty sets is nonempty as having no fixed > meaning, so I'm going to turn around and claim that (*) has no fixed > meaning as far as I can see. It means one thing if (*) is interpreted > in some model of ZFC where the integers are nonstandard and it means > something else if it is interpreted in a model of ZFC in which the > integers are the standard integers. (If there are nonstandard integers > then there are nonstandard proofs, and consistency means that there > aren't any proofs of a contradiction, not even nonstandard ones.) Fair enough. I was unclear. 'cid 'ooh === Subject: Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false? > If you wanted to be really clever, you could have asked why I think > ZFC is consistent iff ZFC has models when the proof of that fact > requires AC instead of this. The proof does not require AC. === Subject: Re: Bowling Score > Puzzle: > If, each frame, you get a strike with probability x, and get a spare > with probability s, and get an open frame with probability (1-x-s), > what is your average score and standard deviation? (Let's assume that > you get a 9 if you don't get a strike on the first ball.) I trust you're also assuming that frames are independent of each other, which doesn't seem to be the case at the highest levels: http://www.economics.pomona.edu/GarySmith/bowling/bowling.html -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity About 1/n, 1/N, 1/n for n E N, inductively for m > n there exists 1/n > 1/m > 0, for no n does there not exist n+1 thus that 1/n <= 1/(n+1). In the limit as n diverges 1/n = 0, and for any finite integer n: 1/n > 0. Where that's agreed upon, there is this notion that the reals are points on a line, beads on a string. As discrete elements if you consider the reals to be a contiguous sequence of points between zero and one, then somewhere between zero and one in the normal progression a single point is the first element of the reals of that interval to satisfy 1/n for some finite positive integer n. No particular definite value of the reals can be considered to be that value, but some indefinite element, an element of the contiguous, totally normally ordered sequence of reals is that value, because the set of all real numbers includes all subsets of the real numbers. An obvious problem with that is that in the normal ordering if you did select any finite integer n then there would be infinitely many values of the sequence preceeding it, where the sequence is not to be doubly infinite, that being an opposite of semi-infinite. So a problem with, or reason to abandon the notion that, the numbers can be defined in that way is that it implies the infinite set of finite integers contains an infinite element, which while conceptually feasible in terms of ubiquitous naturals and the set of all sets containing itself, is still not a clear issue. In that sense the normal ordering progression could contain only indefinite values, with some notion that they were the same numbers as the set of definite values. That is similar to the consideration of the open interval only containing indefinite values: (iota, 2iota, ...), except that as iota could not be rational, as one is the least positive integer the first element of the sequence 1/n would be the least positive rational instead of iota, the first positive real. Here my goal is to avoid confusion and crystallize points of dissent, towards reconciliation and a broader firmament. In the consideration of dissent on points, the unit interval of reals, totally, contains everywhere reals. The notion that there exists for each real number x a real number y greater than x and less than every other real, except for the upper boundary, that the reals are contiguous, sequential, as points besides being continuous as the real number line, is fraught with vagaries on the borderlands of sound, or rather, practiced, definitions. So, of the reals, I consider that there is not a next after zero, and also that there is. Where there is some least positive (infinitesimal transcendental) real called iota, and integral multiples of iota are representative of the real numbers and necessarily not translatable to definite real numbers, except in terms of scalar infinities (x/2x = 1/2), then the set of reals is a contiguous point set on the real number line, besides being a field. If the reals were defined in that way, perhaps obviously in not the standard model, with the multiple definition of rationals and irrationals and as well contiguous points in a sequence, then the normal total ordering would be a well-ordering for any finite interval, where they are not defined in that way, it is not so. Obviously, in standard practice, it is said that it is not so and I'm aware of that. I develop some lines of argument as to what considerations would take place in a redefinition that supports the reasoning otherwise, which is justified for various deductively determined reasons, and consequences of such reasoning. Ross Finlayson === Subject: Re: Infinite number of infinite coin flips > If a diagonal argument worked, it would prove the existence of > cardinals between aleph-0 and c How so? - Tim === Subject: Re: Infinite number of infinite coin flips > What if P and C are both uncountably infinite (say, to the same > degree of aleph-whatever)? Is there a variation of the diagonal > argument which will work in this case? You'd need to define an uncountable number of coin flips more rigorously. For example, label each flipper with an element of some uncountable set X, and also label each coin flip with an element of X. Then the infinite number of coin flips is a function f_x : X -> {0,1}. We can let F be the set of all possible functions of this type. The coin flips actually done can be described by a function B : X -> F. The question is then does there exist B such that B(X) = F. The corresponding diagonal argument is then: Suppose such a B exists. That is, for all f in F, there exists x in X such that B(x) = f. Define f:X->{0,1} by f(x) = 1-B(x)(x). For all x in X, we know that f(x) != B(x)(x), and so B(x) != f. Hence f is not in B(X), which contradicts the assumption. So yes, it does look like the diagonal argument works just fine for uncountable infinities of the same cardinality. - Tim === Subject: Re: Surrogate factoring, update on research <41eecbe4@dnews.tpgi.com.au> Your the professional and the stage is yours - show the world your > magnificence by factoring a few of the RSA challenge numbers, or given that > for some numbers the method does not work, just one of the numbers. Or > maybe you could do a 128 bit number like TSD has already given you? How > about any other RSA type 128 bit number? If I had it to the point where it could factor an RSA challenge number I wouldn't be talking about it, so yes, you are a big dumb-ass for missing the obvious. > No? I think were done. You could have saved a lot of time by simply ignoring my post, or after reading it deciding that you had nothing of value to contribute or cared to contribute so you could say nothing. Instead you babbled nonsense. No, I can't yet factor an RSA challenge number. If I could I damn well wouldn't be freaking talking about the method publicly to give to knuckleheads like yourself. I CAN factor some numbers with a method not in the textbooks. I DO have a theory which strongly suggests that once all the details are worked out that RSA challenge numbers and bigger can be factored in polynomial time. Now I can talk about it publicly at its current state as people like yourself are too freaking dumb to be able to move the research forward, so I'm not worried about you, at all, except to vent a little steam. You're too stupid to advance the basic research, fine. You do not have to proclaim your stupidity to the world by making dumb requests like I should freaking factor a goddamn RSA challenge number as goddamn it, if I could do that I wouldn't freaking be talking about this publicly. As for any other numbers people put up for me to factor, why should I? If you do not think there's anything here, go away. If you're curious you can see what I can and cannot factor easily enough. But I'm not playing footsie with the buttheads among you who refuse to act like civilized people and instead wish to be mean little children in a very public place. Just because you do not see the thousands of people reading these posts it does not mean you look any less freaking stupid. If you have nothing of value to add to this discussion then while you have the right to post I have the right to talk about what a dumb-ass you are as this research can be frustrating and I appreciate that some of you will step up for me to dump all over your stupid dumb asses. The nice thing about Usenet is some other pissant will be here later for me to dump on when I need to vent again. You people come out of the woodwork like cockroaches. James Harris === Subject: Re: enumerable, denumerable, countable? > [second attempt at posting correctly] > This is not a question about the meaning of the three notions but > rather on the usage of the English words for them. > I think I understand the usage of countable and of recursively > enumerable. Now the questions: > Is the word enumerable used in other contexts than recursively > enumerable? In which contexts is the words denumerable used? Denumerable often is used for countable and infinite (and countable almost always includes finite).If I saw enumerable I would guess it meant countable .I have also seen denumerably infinite for denumerable and infinite and I would not be suprised if someone said denumerable as a synonym for countable. === Subject: Re: Minimum of multi-dimensional functions > The function you mentioned will give a spike, but how do i know > where this spike is to ensure my program gives the correct answer? It is centered on (u0,v0...z0), so you choose values for these. > What exactly is u and u0, etc? u is the parameter you labelled in your post. u0 is a particular value you can choose by whatever method you wish. It's a check to see whether your program yielded the correct minimum, or at least close to it. > Also, should your function: > b/(1+sqrt((u-u0)^2+...+(z-z0)^2/c) > have a ')' after the last 2 so the ((u-u0)^2 +...+ (z-z0)^2) is > square rooted, and then the answer to that divided by c? Yes, sorry about that. - Tim === Subject: Re: Primitive Pythagorean triangles of the same area > Gerry, > Can you please provide other known triples > (or links to sites - > I've searched net some but failed). > Also Guy's book is not available to me :=( 77, 38; 78, 55; 138, 5. 1610, 869; 2002, 1817; 2622, 143. 2035, 266; 3306, 61; 3422, 55. 2201, 1166; 2438, 2035; 3565, 198. 7238, 2465; 9077, 1122; 10434, 731. (For anyone coming in late, each pair a, b expands to a primitive Pythagorean triple, 2ab, a^2 - b^2, a^2 + b^2; the three in each line above have the same value for 2ab(a^2 - b^2) [hence, the same area when interpreted as right triangles]; these are the five triples of triples listed in D21 of Guy's Unsolved Problems in Number Theory) -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Factoring problem, solved That is a bold claim to add to previous bold claims of mine, > NOTE: read false for bold. > but here > in THIS post I'll give the underlying theory as it's very basic, and > direct you to a site where you can download a rough, and somewhat > flawed, but still good enough to show the idea, prototype factoring > program, which implements some of the theory. > If you are an American citizen and have solved this problem you are duty > bound to notify the NSA and CIA. These agencies depend heavily on solutions > to this problem for national security. If you do not make every effort to > provide them with your discovery, foreign powers may crush us all. I already mentioned I notified the US Government. I doubt they take me seriously. Maybe they checked on the Internet and saw a lot of pages claiming I'm a crank, eh? In any event, that's neither here nor there. I'm not just making a claim, I'm developing. The code I've put out already is rather oddly powerful, though it's quirky, as what will it factor, and what will it not? I don't know. You can toss some huge number into it, and see, but I prefer to work out the theory rather than waste freaking time. What *amazes* me is that I'm talking mathematics and if you wish to show you are really brilliant, shoot what I have down with mathematics. Stop the yammering you dimwits. I'm trying to concentrate. James Harris === Subject: Re: Factoring problem, solved > Stop the yammering you dimwits. I'm trying to concentrate. > James Harris That's not us, those are the voices in your head. === Subject: Re: Factoring problem, solved > I am increasingly certain that I've solved the factoring problem. You really, really want that to be true, don't you? > I have demonstration code, which is rough as I threw something > together, to test out my own theory, and hopefully to be more > convincing. That code implements a VERSION of my own theory, as I > don't factor both j and T, but only T, as j gets background factored by > the mathematics. While playing with the program, I've noticed that T is always much greater than M, the number the program is asked to factor. Since the program must first factor T, what reason is there to believe that doing so is going to be any easier than factoring M? Also, I don't understand why you expect anyone to be convinced by a program that only works on some numbers. After all, it's very easy to write very small programs which, given time, can factor any integer. And since such programs don't first have to factor a much greater number T, they will probably run much more quickly than your program. === Subject: Re: Alephs - Alephs > Okay, is the difference between aleph 1 and 2 aleph 1? > What about 1 and 4? Generally one can't subtract alephs, but aleph_2 = aleph_2 + aleph_1 so I suppose one could write aleph_2 - aleph_1 = aleph_2. Likewise aleph_4 = aleph_4 + aleph_1 so I suppose one could write aleph_4 - aleph_1 = aleph_4.