Obections to Cantor's Theory (Wikipedia article)
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From: Alan Morgan on 20 Jul 2005 19:24 In article <MPG.1d485ba7151dad6989f4e(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >Daryl McCullough said: >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: >> >> >>> There is nothing in Peano's axioms that states explicitly that all >> >>> natural numbers are finite. >> >> Let's get more specific, and consider the sets S_n = the set of all >> natural numbers less than n. Are you claiming that there is a natural >> number n such that S_n is not finite? >Actually, no, I am saying that for all finite n, S_n is finite, and that, if >all n in N are finite, then so is N. Conversely, since S_n is finite for finite >n, if S_n is infinite then n is infinite. >> >> What definition of finite are you using? >Less than any infinite number. And the definition of infinite number is.... greater than any finite number? How do I prove that, for example, 7 is finite? The only way I can do it, with your definition, is to show that it is less than any infinite number. However, I don't know any infinite numbers and I don't know how to compare 7 with them even if I had them. Worse, your definition is consistent with a system where 8 is infinite. Wierd, but perfectly consistent (obviously 9, 10, etc can't be finite, but that's no skin off my nose). Methinks your definition needs some work. Alan -- Defendit numerus
From: Daryl McCullough on 20 Jul 2005 19:12 Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >> "Bigger" in the sense of no surjection from the "smaller set to the >> "larger", is one thing, "bigger" in the sense of having the "smaller" >> set as a proper subset is different. While these two measures happen to >> coincide for finite sets, they do not coincide for infinite sets, as the >> definition of infinite for sets should hint to you. >> >gee, they coincide for finite sets AND infinite sets, under Bigulosity >but I don't suppose you consider that extra consistency any sort of >progress. No, that's no progress at all. You can prove that for finite sets if there is a bijection between set A and set B, then they have the same Bigulosity. But that fails for infinite sets. So Bigulosity is an inconsistent notion of "size". -- Daryl McCullough Ithaca, NY
From: Jeffrey Ketland on 20 Jul 2005 19:31 Daryl McCullough > When I say "There are infinitely many natural numbers" I > mean exactly "There exists a function f from naturals > to a subset of the naturals". You must have mistyped, Daryl. Any finite set has this property. If X = {a, b}, with a=/=b, then define f(a) = f(b) = a, which is a function from X to a subset (indeed, a proper subset) of X, though X is finite. You mean the usual definition of Dedekind-infinity, "X is Dedekind-infinite" means "There is an *injection* f from X to a *proper* subset of X" (or, equivalently, "X is equinumerous with a proper subset of X"). --- Jeff
From: Chris Menzel on 20 Jul 2005 19:18 On Wed, 20 Jul 2005 18:30:08 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > Daryl McCullough said: >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: >> >> >So, inductive proof does not rely on proving for n+1 based on n? The infinite >> >number of successive naturals for which you prove your property do not >> >constitute an infinite number of implied steps in inductive proof? >> >> As has been pointed out before, an inductive proof does not >> have an infinite number of steps. To prove >> "for all natural numbers x, Phi(x)", you only >> need to prove the following two statements: >> >> 1. Phi(0). >> 2. for all natural numbers x, Phi(x) implies Phi(x+1). >> >> You seem to be thinking that proving statement 2 somehow >> requires an infinite number of steps. If that's the case, >> then statement 2 doesn't *have* a proof (because proofs >> have to be finite). >> >> A proof of a universal statement is not the concatenation >> of infinitely many singular proofs. >> > It seems to me you are all being dense and objecting to something I am > not saying. I hope that's not the case. > > The proof has a finite form, much like a recursive algorithm. A > recursive algorithm will run forever if it doesn't have some stop > condition, like running out of nodes in a tree path, which is bad for > a computer program. But this recursive proof, like the recursive > definition of the naturals, DOES go on forever. Your #2 above is the > recursive part of the proof; it proves something true based on the > truth of its predecessor. As with set theory, so with proof theory -- you don't have any idea what a proof is in mathematical logic. Rather, you are using the word "proof" but filling it full of your own meaning. Now that's all fine and dandy, but the problem is that you think because you are using some of the words mathematicians use that you are talking about the same concepts. (The definition is really pretty clear and simple, you should try to learn about it.) > I hope this clears up what seemed rather obvious to me, Yes, indeed, it is very clear that you are talking about your made up notion of proof (among others), and not the mathematical notion. In fact, what you say may actually *be* obvious for the concept of proof that you have in mind. But of course no one will ever know until you try to do some real mathematics and supply a rigorous definition. Until then, you might consider calling what you mean by a proof a "troof" or something -- it is intellectually dishonest to use language that is part of a large community but infuse it with different meanings. Chris Menzel
From: Chris Menzel on 20 Jul 2005 19:20
On 20 Jul 2005 11:51:30 -0700, Daryl McCullough <stevendaryl3016(a)yahoo.com> said: > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: >>imaginatorium(a)despammed.com said: >>> Tony has no clue what mathematics is, nor how it is done, so he >>> doesn't normally bother with definitions. The closest we got from >>> him for a definition of "finite" was that a finite number is less >>> than an infinite one. And you can guess the "definition" of >>> infinite. >>Well, that's about as close to a lie as one can get, eh? > >>I asked for a definition of infinite, and no one could give me a >>definition of that word. The best I could get was that an infinite >>set can have a bijection with a proper subset, which is hardly a >>definition of the word "infinite". > > On the contrary, that's a perfectly good definition of the concept > "infinite set". (assuming choice ;-) |