Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 27 Jul 2005 23:11 In article <MPG.1d51a39044034e06989fbb(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > You are being challenged on the notion of 'infinite natural > > numbers' as necessarily existing in the set of all standard natural > > numbers, because that's poppycock. > Okay, given the standard Peano axioms, it would seem that one might > not be able to count to infinite numbers, because you can't define > the point where they beocme infinite. > But any infinite set of whole > numbers MUST contain infinite values for the reasons I have put > forth. The reasons TO puts forth are delusional. They require the existence of a finite ordered set with no largest member, and a finite set whose ennumeration is not possible. > Either one abandons the idea that the set of naturals is > infinite, or one includes infinite numbers in the set. Or one faces reality for a change. Finite ordered sets have largest members in any standard axiom system for the naturals. Until TO produces some workable aternative to the Peano properties, he is SOL.
From: Poker Joker on 27 Jul 2005 23:15 "Daryl McCullough" <stevendaryl3016(a)yahoo.com> wrote in message news:dc9h1k08iu(a)drn.newsguy.com... > Tony Orlow (aeo6) wrote: > >>> Who cares about S_oo? That's not one of the S_n. > >>You mean there aren't an infinite number of n in N? > > Okay, maybe this will make it clear to you: There is > exactly 1 natural number that is less than 1: namely 0. > There is exactly 2 natural numbers that are less than 2: > namely 0 and 1. There are exactly 3 natural numbers that > are less than 3: 0, 1, and 2. > > You see the pattern? For any n, the number of naturals > less than n is equal to n. > > Now, how many natural numbers are less than infinity? I disagree with grouping infinity in with the natural numbers. That's what Cantorians do.
From: Virgil on 27 Jul 2005 23:17 In article <MPG.1d51a48eba0be396989fbc(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > 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. > > > > -- > > Daryl McCullough > > Ithaca, NY > > > > > That's what they say, and I disagree. The "largest element" is not a valid > criterion for finiteness. Why not? It is certainly compatible with the Cantor criterion for finiteness. TO has yet to demonstrate that it violates any rule of standard arithmetic, or give any example not directly related to his claim for the naturals. On the contrary, if 1 is finite and n finite implies n+1 finite, one can prove it is a requirement for finiteness.
From: Virgil on 27 Jul 2005 23:19 In article <MPG.1d51a5013dbc0344989fbd(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > malbrain(a)yahoo.com said: > > Tony Orlow (aeo6) wrote: > > > > > In any case, sure, the program will spit out finite numbers, snce > > > it is a finite machine running in finite time. If the machine had > > > infinite capacity and infinite funtime, it could conceivably > > > produce infinite results. > > > > In order to finish generating the natural numbers in 1 second, it > > would need to spit out natural number n with a calculation time of > > 1/2^n. I believe George first noted this fact in a post to Daryl a > > few years ago. > > > > Unfortunately, I've never found a way to configure my > > java-virtual-machine to accelerate with the number of steps taken, > > so I've not found much use for his observation. karl m > > > > > I think it might help to perform the calculation while plummeting > into a black hole Is TO volunteering?
From: Virgil on 27 Jul 2005 23:33
In article <MPG.1d51a6ad3b758230989fbe(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Virgil said: > > In article <MPG.1d506ab0de98a030989f9e(a)newsstand.cit.cornell.edu>, > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > 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. > > > > Or they may not. > > > > > > > how do you have an infinite set of strings with only finite lengths? > > > > The usual way, by not having any finite bound on their lengths. > Only an infinite bound? Then they CAN be infinite? You make no sense. > > > > > > > > > > >I am not assuming anything except for this fact. > > > > > > > > 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, then S^L CANNOT be > > > infinite > > > > No one is requiring any S^L to be infinite, we merely deny that there is > > any finite lid on the size the S^L can attain, as S and L are allowed to > > increase without finite limit. > Oh, so now you are not requiring the set of naturals to be infinite? I could > have sworn..... Read what I said TO! I am still rejecting any finite limit on the size of the set of finite naturals. Does TO wish to propose some specific finite limit on the size of the set of finite naturals? > > > > > For finite S, > > > S^L can ONLY be infinite with infinite L. Why is this so hard to > > > understand? If S and L are both finite, then S^L is finite, isn't it? > > > > But no one is claiming that any of S, L or S^L is infinite, we just say > > that any of them can be larger that any finite natural you choose to > > name. that is, there is no finite limit on their sizes. > So S^L, the size of the set, is finite but unbounded? Okay. TO again attempts to mislead. What does he mean by "the set"? There are infinitely many finite S^L's of various sizes for infinitely many different sets of finite strings with infinitely many different finite S's and infinitely many different finite L's, but TO tries to imply that there is only one? > > Why TO rails against something that no one is claiming is because he has > > no valid argument by which to refute what we actually are claiming. > I believe, and correct me if I am wrong, that you were ALL claiming > the set of finite natural numbers was infinite. I COULD be > mistaken.... TO gets one thing right for a miracle, but re S^L, which was the issue in question, we were not claiming anything. What we arre refuting is TO's slight of hand argument that S^L is in any way relevant to whether there are finitely or infinitely many finite naturals. |