Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 25 Jul 2005 15:08 Daryl McCullough said: > 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 > > Huh? Bigulosity doesn't rely on bijections. That why it DOES work. -- Smiles, Tony
From: Virgil on 25 Jul 2005 15:09 In article <MPG.1d4eb5ec6ddf7a0c989f68(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > 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? Because in ZF and ZFC and other similar axiom systems, the set of naturals is well-ordered, meaning that every non-empty subset contains a smallest member, so the set of n in N for which f(n) is 'missed' is either empty or has a smallest (first) member. > > > > 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. TO must at least prove that such a value exists in order to successfully claim one exists, even if he cannot give a value for it. So far TO has only claimed, and not proved, that any such number exists. Considering TO's track record on claims, that's just not good enough.
From: David Kastrup on 25 Jul 2005 15:11 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > Daryl McCullough said: >> > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: >> >> >> 1. Phi(0). >> >> 2. for all natural numbers x, Phi(x) implies Phi(x+1). >> >> >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 unlike an algorithm, there is no implied infinite number of >> steps. > Yes there is. You show the proprty true for n=0, then for succ(n), > then succ (succ(n)), etc. This is the justification for the > axiom. It may be the _motivation_ for the axiom. But the axiom's consequences are independent from its motivation. > That is why inductive proof is said to work. That is why the axiom was chosen: to have a better mechanism than needing to check each case individually. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Tony Orlow on 25 Jul 2005 15:14 Chris Menzel said: > On Wed, 20 Jul 2005 20:55:58 +0100, Robert Low <mtx014(a)coventry.ac.uk> > said: > > Daryl McCullough wrote: > >> That's not true. If S is an infinite set of strings, then there is a > >> difference between (1) There is no finite bound on the lengths of > >> strings in S. (2) There is a string in S that is infinite. > > > > Except that TO claims that (1) implies (2), though I can't > > even get far enough into his head to see why he thinks it, > > never mind finding his 'argument' convincing. > > This seems to be a fairly common element in crankitude. I've seen > several folks argue here and elsewhere that there can be infinitely many > natural numbers only if there is an infinite natural number. (Indeed, I > think TO believes this, as I believe I saw reference to an "infinite > natural" in one of his posts.) The origin of this idea sometimes seems > to reside in imagination -- the poor afflicted fellows picture the > number sequence as something like an endless string of beads that > eventually disappears to nothing. There is thus no perceptual > difference between *really really long* proper initial segments of the > string and the entire string itself. So the entire string is the same > sort of thing as its really really long proper initial segments. Other > times, there seems to be some sort of a priori cardinality principle at > work: for every set of natural numbers there is a natural number that > numbers them. No finite natural number numbers all the finite natural > numbers, so (obviously) there is an infinite natural number. > > Whatever. Kinda sad. > > What is sad is that you say you have heard the same arguments repeatedly, and still don't seem to understand them. If there is a constant finite difference of one between any two consecutive natural numbers, and there are an infinite number of them, then indeed the overall range of values becomes infinite. You might want to try that imagination thing you were just making fun of. Maybe you can rediscover the chemical structure of benzene. -- Smiles, Tony
From: georgie on 25 Jul 2005 15:15
MoeBlee wrote: >Axioms are rigorous in that there is an effective method by which to >determine whether a formula is an axiom. Please provide an effective method by which we may determine whether a formula is an axiom. |