Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 25 Jul 2005 09:40 Virgil said: > In article <MPG.1d483916d7024f5b989f44(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > Virgil said: > > > In article <MPG.1d4725fbdfa9bb24989f2d(a)newsstand.cit.cornell.edu>, > > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > > > > > > > There is a simple, demonstrably valid proof of Cantor's Theorem > > > > > in ZF set theory. So you must think the proof is unsound. > > > > > Which axiom of ZF do you believe to be false? > > > > > > > > > > Chris Menzel > > > > > > > > > > > > > > I was asked that before, and never got around to fully analyzing > > > > the axioms for lack of time, but the diagonal proof suffers from > > > > the fatal flaw of assuming that the diaginal traversal actually > > > > covers all the numbers in the list. > > > > > > If it misses any of them, it must miss a first one. Which is the > > > first one misssed? > > > > > > If there is not a first one missed, then none are missed. > > > > > The first one missed is the first one directly below the diagonal > > traversal. > > > If any are missed then there must be an n in N corresponding to that > "first one" missed. Why? > > If TO cannot produce such a finite n from N, then no such n exists, and > that anti-diagonal is complete. That is non-logic. You might as well say fsirnlskrnlsnglk QED. > -- Smiles, Tony
From: Randy Poe on 25 Jul 2005 09:45 Tony Orlow (aeo6) wrote: > The fact is that any set of whole numbers Correction: any finite set of whole numbers > greater than or equal to 1 MUST have > at least one element with a value at least equal to the size of the set. That > is a fact for finite sets > and has been proven. by induction on finite sets. - Randy
From: Tony Orlow on 25 Jul 2005 09:49 Virgil said: > In article <MPG.1d4839e77d7d2c0a989f45(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > Virgil said: > > > > If TO's assumprtions were actually the case, there would have to be a > > > finite natural so large that adding 1 to it would produce an infinite > > > natural. But TO cannot produce either a largest finite nor a smallest > > > infinite, so the set of all finite naturals is already big enough. > > > > > We have been through all this before. You lay these requirement on me, but > > when > > I say you cannot have a smallest infinite omega, or omega-1 would be finite, > > you say omega-1=omega. > > I do not say anything at all about omega. I merely go by the Peano > properties, which prohibit TO's vision of "infinite naturals". > > > So, I can play your stupid game, and say that alpha is > > the largest finite, but alpha+1=alpha. Tada! I am as senseless as you! Isn't > > that great? Let me know when you want to get off the "largest finite" kick. I > > won't be responding to it any more. > > It is not me who keeps requiring a largest finite, but TO, by his > delusional insistence that there must be non-finite naturals within the > Peano system. > While it may seem that counting ala Peano can never produce any infinite numbers, the insistence on the set being infinite requires the existence of infinite values in the set. Perhaps you should call your set "boundless" or "bigfinite" or something, but it's not "infinite" if you only count a finite number of times and only have finite values. You know, the Peano axioms can easily be inverted to produce infinite whole numbers that count down, with exact symmetry compared to the finite end of the number circle. If one uses such a set of axioms, does that mean that finite whole numbers cannot exist, because that set of axioms doesn't seem to allow us to count down that far? This insistence on determining the dividing point between finite and infinite is clearly a waste of time, and your repetitions of this non-point don't make it any more important. You can't count through the divide in any finite number of steps. So what? That doesn't mean that one side exists and the other doesn't. Infinite set sizes ARE infinite whole numbers. I can't understand why this isn't clear to everyone. -- Smiles, Tony
From: Tony Orlow on 25 Jul 2005 09:52 stephen(a)nomail.com said: > In sci.math Daryl McCullough <stevendaryl3016(a)yahoo.com> wrote: > > 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". > > These seems to be another common misconception among > the anti-Cantorians that words cannot have specific > meanings in specific contexts. Somehow they > think an all-encompassing definition of 'infinite' must > be provided before someone can say what an infinite set is. > I am not sure what they mental hangup is. I wonder > how any of them would ever learn a foreign language. > > What is so hard about > "A set is infinite if there exists a bijection between > the set and a proper subset of the set." > ? There is no need to get all mystical and metaphysical > just because the word 'infinite' shows up. The word 'infinite' > may have other definitions in other contexts, but that is > true of all words, and is irrelevant in a discussion of > infinite sets. > > Stephen > The problem comes whenever someone wants to use the word "infinite" in any other context. There ARE no infinite numbers, only infinite sets, in your lexicon. Talk about needing to learn about language. What is so hard about infinite numbers, or trying to define the word itself, independent of bijection genuflection? -- Smiles, Tony
From: Robert Low on 25 Jul 2005 09:56
Tony Orlow (aeo6) wrote: > The problem comes whenever someone wants to use the word "infinite" in any > other context. The problem comes when people use a word in one context as if its meaning were specified by a different context. That's what an amphiboly is. > There ARE no infinite numbers, only infinite sets, in your > lexicon. Not only do we know about infinite numbers, but there are at least two different kinds of them: cardinal and ordinal ones. (Still other notions of infinite number occur in other contexts.) Again, clear definition of what is meant in a particular context is required. > Talk about needing to learn about language. What is so hard about > infinite numbers, or trying to define the word itself, independent of bijection > genuflection? What is so hard about the idea of context sensitivity, and accepting that the word 'infinite' can mean different things in different contexts? Does it bother you than 'bread' can refer to products made from different grains, or can even refer to 'money', depending on the context? |