Obections to Cantor's Theory (Wikipedia article)
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From: imaginatorium on 20 Jul 2005 14:22 Tony Orlow (aeo6) 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. Not infinite. With a known end, or bound. I am > sure you know what I mean. Hmm. Been here before... Does "with a known end" mean that you can name the known end of the set of pofnats, which you tell us is finite? And tell us what its successor is? And explain how exactly there is no contradiction with the Peano axioms? (For anyone who hasn't been following this: pofnats are the (plain old finite) normal mathematical naturals; the Tonats are the Orlovian naturals, which look a bit like the n-adics.) I don't suppose Tony that in your recess you've considered how you're going to learn abstract algebra, and recreate it? Since you don't know any set theory yet, it's going to be a big job... Brian Chandler http://imaginatorium.org > > > > -- > > Daryl McCullough > > Ithaca, NY > > > > > By the way, I don't suppose you would be going to GrassRoots this weekend? > -- > Smiles, > > Tony
From: Virgil on 20 Jul 2005 14:26 In article <MPG.1d4830d09199666c989f3e(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > stephen(a)nomail.com said: > > In sci.math Barb Knox <see(a)sig.below> wrote: > > > In article <MPG.1d4726e11766660c989f2f(a)newsstand.cit.cornell.edu>, > > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > [snip] > > > > >>Infinite whole numbers are required for an infinite set of whole numbers. > > > > > Good grief -- shake the anti-Cantorian tree a little and out drops a > > > Phillite. Here's a clue: ALL whole numbers are finite. Here's a > > > (2nd-order) proof outline, using mathematical induction (which I > > > assume/hope you accept): > > > 0 is finite. > > > If k is finite then k+1 is finite. > > > Therefore all natural numbers are finite. > > > > Talking to Tony is a waste of time. He does not understand > > induction and is a firm believer in "after infinity". He > > is a fine example of the non-mathematical sort who complains > > about Cantor. > > > > Stephen > > > Nice ad hominem. You never understood any of my points. Take your fingers out > of your ears and stop with the "blah blah" Why should one even bother to listen to one who continually makes no sense, and continually refuses to listen to what does make sense.? TO's "points" have all been refuted. For example, Barb Knox, above, shows succinctly why TO's "infinite naturals" are a delusion.
From: Daryl McCullough on 20 Jul 2005 14:10 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. -- Daryl McCullough Ithaca, NY
From: Virgil on 20 Jul 2005 14:35 In article <MPG.1d4832e1f03a8d50989f3f(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Virgil said: > > In article <MPG.1d4712ec2f75d957989f26(a)newsstand.cit.cornell.edu>, > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > Is this the > > > same as the proof concerning the "uncountability" of the reals? > > > > Quite similar, but not identical. The basis is the same: showing that > > any mapping from the smaller to the larger _must_ fail to be surjective. > > > And why is a lrger set necessarily uncountable, or unenumerable? The DEFINITION of a set being "non-denumerable" or "uncountable" is that there be no surjection from N to that set. So that whenever one can prove that there is no such surjection to a set, that set is, BY DEFINITION "non-denumerable" and "uncountable". If you can > draw bijections between naturals and evens and declare them the same, when > one > is obviously twice the size of the other, then why can't a similar bijection > be > created. After all, wouldn't you say that the set of all integral powers of 2 > is a countable set? Or all log2's of natural numbers? In my book, there are > 2^N > log2's of natural numbers. That doesn't make it uncountable. It just makes it > a > bigger set. "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.
From: Tony Orlow on 20 Jul 2005 14:38
imaginatorium(a)despammed.com said: > > > Daryl McCullough wrote: > > 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? > > > > What definition of finite are you using? > > 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". In fact I went to the etymology, which literally means "without end". Finite means with a known end or bound, and infinite means without end. Of course, I got all sorts of flack for my definition, from those that couldn't even suggest one outside of the set theory they were regurgitating. let's try to be straight here, and no more insulting than necessary, so it doesn't come back to bite us, why don't we? > > As best I can grasp it, the central principle of Orlovian pseudo-maths > is that "infinite numbers", never clearly being defined, but reached by > continuing from finite numbers through a "twilight zone" (whose > existence is anecdotally stated), have essentially the same sort of > properties as "ordinary numbers". The Orlow-refutation of the diagonal > proof rests on the "fact" that a list of infinite sequences of digits > is not a quarter-plane, as one might imagine, but a rectangle, of width > P and height Q (P and Q being some Orlovian infinite numbers), so of > course the diagonal hits the (infinite) side at some point. The > mathematical concept of a sequence being endless means (to us) that > there is no end, but that doesn't stop Tony using the end to prove > something. You'll notice he gets a bit irritable when people point out > that one of his "proofs" doesn't work because there *isn't* a largest > integer (or whatever). (sigh) Yes you are on a bullshit roll here, Brian. At least I offered a definition for infinite, which none of you did. The "twilight zone" we discussed is that impassable zone between finite and infinite, where your impossible largest finite and you fictitious omega meet, which I repeatedly agreed could not be transcended through finite addition or incrementation. I asserted that many of the same properties hold on either side of this zone, in mirror image, which I still contend is true. You assessment of my objection to the diagonal proof is essentially correct, and I still stand by it. Yes I get sick of the "largest finite" mantra, especially in the context of your equally impossible omega, your smallest infinite, which you only maintain through the use of "non-standard" arithmetic where omega-1=omega. I offered three proofs regarding the naturals, only one of which had anything to do with a largest member. One of the others was "refuted" by saying that induction doesn't prove things for an infinite set (bullshit), 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. The one using digital representations has not been refuted at all, but largely ignored, since you CANNOT have an infinite number of digital numbers, or strings on any finite alphabet, without allowing infinitely long strings. That's how it went, for the record. What really "irritates" me is deliberate bullshit, and lies regarding what I have said. I do not need people summarizing my position, thank you. I can do that very well myself. > > Tony has any number* of "proofs" that an infinite set of natural > numbers must include "infinite naturals", but these are generally > circular. The one from "information theory" says that since there can > only be a finite number of strings of finite length (even if the length > has no limit), then to get an infinite set of numbers, you must include > some that are infinitely long. The bit after "since" is a restatement > of what he purports to prove, but he ignores people pointing this out. Ahem! That is another misrepresentation. The bit after "since" is a statement about symbolic systems, and is a fact outside of the natural numbers. Given a set of symbols of size S, one can construct a set of all strings of length L, and the set of strings has size S^L. This is a fact, which when combined with the fact that digital strings are strings on a finite alphabet (S is finite), S^L can only be infinite if L is infinite. Therefore, an infinite set of digital numbers MUST contain numbers with infinite numbers of digits. If there are infinite numbers of significant digits to the left of the digital point, as would be the case with infinitely long whole numbers, then by the definition of digital systems, such strings represent infinite values. Refute that, specifically. > > * He claims three, but since anything follows from False, surely such > "proofs" can be generated without limit. Would you like some ketchup with that foot? > > Brian Chandler > http://imaginatorium.org > > -- Smiles, Tony |