Obections to Cantor's Theory (Wikipedia article)
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From: stephen on 26 Jul 2005 19:07 In sci.math Daryl McCullough <stevendaryl3016(a)yahoo.com> wrote: > Tony Orlow (aeo6) wrote: >> >>Daryl McCullough said: >>> But for the set we are talking about, there *is* no L. We're talking >>> about the set of *all* finite strings. That's an infinite union: If >>> A_n = the set of all strings of length n, then the set of all possible >>> finite strings is the set >>> >>> A = union of all A_n >>> = { s | for some natural number n, s is in A_n } >>> >>> This set has strings of all possible lengths. So there is no L >>> such that size(A) = S^L. >>If those lengths cannot be infinite, then the set cannot be either. > Why do you believe that? >>Either you have an upper bound or you do not, and if there is no >>upper bound on the values of the members, then they may be infinite. > Why do you believe that? >>> You are assuming that every set of strings has a natural number L >>> such that every string has length L or less. That's false. >> >>I am saying that if L CANNOT be infinite > I'm saying that there *is* no L. So don't talk about the case > where L is infinite or the case where L is finite. I'm talking > about the case where there *is* no maximum size L. > Why do you think that there is a maximum size L? I doubt you will get any rational response. The idea that a set of finite objects must be finite is so engrained in some people's mind that they cannot see past it, despite all the obvious contradictions. For example, presumably there is some maximum length to the finite binary strings, which we will call L. How many binary strings are there then? 1 + 2 + 4 + ... + 2^L, which we all know is 2^(L+1)-1, which is finite, and is clearly larger than L (assuming L > 0). Now why someone would believe that there can exists 2^(L+1)-1 binary strings, but there cannot exist binary strings with length 2^(L+1)-1 is quite beyond me. They are both finite numbers. Why is the limit on finite string lengths smaller than the limit on finite sets of finite strings? Stephen > -- > Daryl McCullough > Ithaca, NY
From: malbrain on 26 Jul 2005 19:13 David Kastrup wrote: > malbrain(a)yahoo.com writes: > > > Tony Orlow (aeo6) wrote: > >> malbrain(a)yahoo.com said: > >> > Virgil wrote: > >> > > In article <MPG.1d4863d52071fde5989f51(a)newsstand.cit.cornell.edu>, > >> > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > >> > > >> > > > and that I was trying to prove > >> > > > things about sets, not numbers, which is also bullshit, since I was > >> > > > proving a property regarding a set DEFINED by a natural number, which > >> > > > is ultimately a property of that number. > >> > > > >> > > But the set N is not defined by any one natural number > >> > > >> > Under Tony's theory, the number representing N is defined by a string > >> > of an infinite number of ones. Yes, more than one Turing machine can > >> > produce this. karl m > >> > > >> > > >> Actually there are two ways to look at it. In unsigned binary, yes, > >> an infinite number of 1's is the largest number possible. Since we > >> start with all 0's representing 0, the size of the set, N, will be > >> one more than 111...111. It will be 000...001:000...000, or one > >> unit infinity. > > > > There's already a STANDARD method for coding real numbers using the > > integers. It's published by the Institute of Electrical and > > Electronic Engineering. > > Uh, no. That's a method for coding floating point numbers. That's > quite something else. Well, it's true that I don't actually use FP beyond introducing decimals to integers. Taking 13 and producing 1.3 with a FP multiplication. To do money, we use cents or mils, as integers, depending on the precision required. That's our connection to REAL NUMBERS. karl m
From: malbrain on 26 Jul 2005 19:18 briggs(a)encompasserve.org wrote: > In article <1122393788.077928.129590(a)o13g2000cwo.googlegroups.com>, malbrain(a)yahoo.com writes: > > Daryl McCullough wrote: > > > >> Nothing in mathematics is excepted without question. Not by > >> mathematicians, anyway. > > > > I think you meant ACCEPTED. See Barb's post for a discussion of > > EXCEPTED. > > > >> Yes, it is certainly the case that *if* you can prove by > >> induction "forall x, Phi(x)", *then* you can write a > >> corresponding recursive function that given a number n, > >> produces a proof of Phi(n). Nobody disputes that. What > >> people are disputing is your bizarre belief that proving > >> "forall x, Phi(x)" by induction means that you have proved > >> Phi(0), Phi(1), Phi(2), ... It means that you *can* prove > >> all those infinitely many statements, not that you *have*. > > > > Sorry, but the axiom states that you HAVE INDEED proved your assertion > > for each and every n when its conditions are satisfied. > > The axiom makes no statement about what you *can* prove. > The axiom makes no statement about what you HAVE INDEED proved. Well, it never hurts to look again. > The axiom states that the _PROPERTY HOLDS_ for each and every n > given that the property holds for 0 and that whenever the property > holds for i, the property holds for S(i). "Three years later Fermat identified an important property of the positive integers, namely that it did not contain an infinite descending sequence. He did this in introducing the method of infinite descent 1659: .... in the cases where ordinary methods given in books prove insufficient for handling such difficult propositions, I have at last found an entirely singular way of dealing with them. I call this method of proving infinite descent ... The method was based on showing that if a proposition was true for some positive integer value n, then it was also true for some positive integer value less than n. Since no infinite descending chain existed in the positive integers such a proof would yield a contradiction. Fermat used his method to prove that there were no positive integer solutions to x4 + y4 = z4. " karl m
From: malbrain on 26 Jul 2005 19:30 David Kastrup wrote: > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > When the only way to form a bijection is with a mapping function, > > then that function needs to be taken into account. This nonsense > > about an infinite set of finite whole numbers is pretty bad too, > > Well, it seemingly is not a concept accessible to everybody, > surprising though this may appear. > > Core at this problem is the inability to differentiate between > infinite (a property of a single element) and arbitrarily large (a > property of available elements from an infinite set), in short, the > inability to distinguish between How is this going to help? You're dealing with pre-axiom of infinity. There are only the Peano axioms. Why don't you try describing the difference between infinte, indetermate, and arbitrarily large??? >From the history of computers: 1/0 = infinite, 0/0 = indefinite and 2^60-1 equals arbitrarily large. karl m
From: malbrain on 26 Jul 2005 19:36
Daryl McCullough wrote: > Tony Orlow (aeo6) wrote: > > >> (3) The answer to "Is there a largest pofnat?" is somehow neither > >>'Yes' nor 'No'. > >No, the answer is no. > > But you claimed that the set of all finite naturals is a finite set. > Every finite set of naturals has a largest element. I would say you're going to have to try this without resorting to sets at all. I don't know if it's possible; but then again, we don't have the inquisition bearing down either. Is the notion of the number of natural numbers the only connection to the axiom of induction? karl m |