Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 20 Jul 2005 23:24 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. Then TO is claiming that N = S_n for some 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? > I am sure you know what I mean. In math, that's not good enough. If you can't give a precise definition, when asked, you don't know what you're talking about.
From: Virgil on 20 Jul 2005 23:29 In article <MPG.1d485c6dc91e5c36989f4f(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > What it really shows is that digital systems with a given number of > digits have more strings than digits. it is not necessary to have > aleph_0 digits, if you allow for smaller infinities. Smaller than what? The set of naturals is as small as infinite sets get. TO seems to be saying that because the set of all naturals has no limit on thet number of digits needed that there must be some one neatural with no limit on the number of digits needed, but it does not follow. It is just another instance of TO's quantifier dyslexia.
From: Virgil on 20 Jul 2005 23:35 In article <MPG.1d485d5b56de5b6989f50(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > And yet, one cannot apply an increment an infinite number of time without > adding infinity. That TO says it cannot be done does not prove that it can't be done. Anyone but TO, and a few others of equally limited capabilities, can do it quite easily.
From: imaginatorium on 20 Jul 2005 23:40 Tony Orlow (aeo6) wrote: > imaginatorium(a)despammed.com said: > > > > > > 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? > The contradiction come from your decalration that they are all finite. Hmm. This is getting tricky. Look, for the purposes of argument, I am accepting the notional existence of your 'set' of 'natural numbers', the Tonats, some of which are infinite, and some of which are finite. I am then *defining* a different set, a subset of the Tonats, which I call the pofnats, and I am defining this set to include only those of the Tonats that are finite. That means that when I say "the pofnats", this expression could be replaced by the expression "the subset of the Tonats containing only those which are unambiguously finite". This replacement is totally impractical, but does not change the meaning at all. So when you say the problem is in my "declaration", I'm not sure what you mean. Can I not select those of the Tonats that are finite? OK, I know what your getout is here: to you being "finite" is something like being "small" or being "interesting". 139 is ever so slightly less finite than 3 is. Well, this is why people keep asking you what you mean. They can't understand your definition of "finite" if it means that some numbers are a bit less finite than others. When people say they can't understand your terminology, please have the decency to understand that this means that they really *can't* understand what you are saying. But this "gradated finitude": really, how does it work? It sounds like you could rephrase "finite" as "close to 0". In some (rather woolly) sense, 10^10^100 is "close to zero", when we think about the interval [10^10^100, 47^47^470]. Yet 13 is clearly closer to zero than 10^10^100. How could this have anything to do with "having an end"? The interval [0, 13] has a left end at 0 and a right end at 13. The interval [10^10^100, 47^47^470] has a left end at 10^10^100 and a right end at 47^47^470. Who but Mueck could think these ends are in any way less endular than any other end? * Sorry, I'm probably using 'interval' nonstandardly. By [n, m], n and m pofnats, m>n, I mean the set {n, n+1, ... m-1, m} So if you want to use "gradated finitude", we really need a definition, or a first draft of a definition. I keep suggesting that a reasonable, if informal, definition of a finite set, that does not rely on any formal set theory, is that a set is finite if it can be counted against a ditty, and the ditty stops. If the ditty does not stop, the set is infinite. Any claim that there can be some half-way house at which the ditty 50% stops is quite beyond my comprehension, without a very large amount of explanation. > I have > already repeatedly pointed out the flaw in that proof. Go ask your infinite- > series friends. They'll tell you. You keep wittering about infinite series, and posting links to a Mathworld page on tests for convergence. The relevance of this is hard to determine. In particular, there is no "proof" at all, in what I just said. I selected a set, the pofnats; you appear to accept that this set exists; you claim it is finite; you suggest this means it has a last member; I ask how this is compatible with the Peano axioms, which say that any pofnat has a successor, since this appears to imply that the successor is also a pofnat. This appears to me to be not a proof, but simply a contradiction, and I ask how you resolve it. You also appear to wish that there are - somewhere in the world of normal maths - people (the "infinite series" crowd) who could explain all these errors in "Cantorianism". Why don't you find one? Try your local college, or whatever. After all, if you are going to be able to "destroy" set theory, you can expect to be at least as famous as JSH appeared to expect to be, and that's Quite Famous. Brian Chandler http://imaginatorium.org
From: Virgil on 21 Jul 2005 00:10
In article <MPG.1d4863d52071fde5989f51(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > 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. Cantor gave a good one: A set is finite IFF there does not exist any injection from that set to any proper subset. it has been presented to TO many times. We still use it. It follows that a natural is finite IFF the set of naturals up to and including it is a finite set by the above definition. > 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". It is precisely a definition of the word "infinite" at least as applied to sets. That TO does not understand it says much about TO's understanding, but nothing against the validity of the definition. > 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? Was that a dictionary of mathematical terms? The mathematical meanings of a large variety of words are quite different that their everyday meanings. > > > > 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. If you can't get there in a finite number of steps, it does not exist. > Yes I get sick of the "largest finite" mantra, especially in the > context of your equally impossible omega, your smallest infinite, Can TO produce a set which is Cantor-smaller than the standard N, but still Cantor-infinite > I offered three proofs regarding the naturals, No one has seen anything yet by TO that qualifies as a proof. > 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 what does induction prove about infinite sets according to the Gospel by TO. > 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 The one using digital > representations has not been refuted at all, but largely ignored, Since it is idiotic on its face, it is best ignored. > since you CANNOT have an infinite number of digital numbers, or > strings on any finite alphabet, without allowing infinitely long > strings. Everyone but TO can quite easily do it. > That's how it went, for the record. What really "irritates" > me is deliberate bullshit, and lies regarding what I have said. That TO calls something BS does not make it so. > I do > not need people summarizing my position, thank you. I can do that > very well myself. Actually, TO does a lousy job of it, since he keeps getting everything wrong. > > > > 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. Unless one puts some limit on the maximum length of such strings there is no limit on lengths, which is TO's definition of infinite, but without requiring any one string to be without limit. |