Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 20 Jul 2005 16:31 Virgil said: > In article <MPG.1d4816be96c6f78d989f36(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > Chris Menzel said: > > > > Pretty clearly, you aren't terribly well-educated in set theory. Don't > > > you think you should understand a field before you try to point out its > > > flaws? And don't you see that it makes you look rather silly? Would > > > you consider attacking superstring theory without understanding basic > > > quantum mechanics? > > > I have not immersed myself deeply in set theory, no. But, I can see clearly > > that some of the conclusions are wrong > > Without knowing how and why those conclusions have been reached? > > What sort of crystal ball are you using? > An infinitely average square black one made of liquid stone, not dissimilar in appearance to your head. -- Smiles, Tony
From: Tony Orlow on 20 Jul 2005 16:32 stephen(a)nomail.com said: > In sci.math Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > stephen(a)nomail.com wrote: > > >> It seems like a lot of the "anti-Cantorians" and other > >> mathematical nay-sayers tend to start with their conclusions, > >> and then try to work backward. The idea of starting > >> with a fixed set of well defined axioms and working > >> forward seems totally alien to them. > > > Nah, nah. First comes the theorem. And then comes the proof. > > > Han de Bruijn > > You cannot prove something without axioms. So if you > start with your conclusions, you have to create some > axioms before you can find a proof. Unfortunately they > never seem to actually define any axioms. Look at Tony Orlow > in this thread for a fine example of this sort of behavior. > > Stephen > I am working on a set of axioms. One of the central ones is N=S^L, which I have used repeatedly. It's difficult work, but it will be done over vacation in a few weeks, hopefully. -- Smiles, Tony
From: Daryl McCullough on 20 Jul 2005 16:12 Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > >Daryl McCullough said: >> Not quite. "Uncountable set" means "set with a larger cardinality >> than the set of naturals". >Yes, that seems to be the Cantorian definition. Does it actually follow >that an infinite set larger than the naturals can't be enumerated? Yes, it certainly does follow. The definition of "enumerated" means "put in one-to-one correspondence with the set of naturals (or a subset of the naturals)", which implies "having the same (or smaller) cardinality as the set of naturals". >Personally, I don't see the connection at all, and view it as a >conflation. I don't see how you could fail to see the connection. >That's interesting, since the mathematiicians here seem to >like to haggle over words that they themselves can't define, >like "infinite". That's false. Nobody likes to haggle over the definition of "infinite". It has a perfectly good definition. >I use Bigulosity, to distinguish my measures from cardinality. Fine. Give a definition of Bigulosity. What *I* mean by a definition is a rewrite rule so that any sentence involving the word "Bigulosity" can be rephrased into an equivalent sentence involving standard mathematical and logical concepts: existential quantification, universal quantification, addition, multiplication, equality, set membership, >Try "unending". Or, as I requested, give ANY synonym, or >definition of the word itself. For mathematical purposes, a word is defined if you are able to rewrite any sentence involving that word into an equivalent sentence involving only standard concepts. >> Nothing of any importance about mathematics would change >> if we substituted different words for the basic concepts. > >Then you shouldn't be having a word problem with me, right? I have no idea what you are talking about when you use the words "infinite", "finite", "larger", "smaller", "size", etc. >My arguments have NOTHING to do with terminology. Then rephrase them without using the terminology "infinite", "finite", "size", "larger", "smaller", "without end" etc. Rephrase it using only *mathematical* concepts. 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". >If I say that the set of evens is smaller than the set of naturals, Don't say that. Say what you mean mathematically. For example, maybe you mean The set of evens is a subset of the set of naturals. I agree with that. But don't use the word "smaller" because that word doesn't mean anything definite. >I think you all know what I mean No, I don't. >Cardinality is supposed to be a measure of set size, >> As a challenge, see if you can express your claims about >> infinite sets, or infinite naturals, or set size, or whatever, >> *without* using the words "infinite", "larger", "size", etc. >Yeah sure, and you describe your fluffy pink flying elephant >without using the words "fluffy", "pink", "elephant" or "flying". If I'm talking about fluffy pink flying elephants, then I'm not talking about mathematics. When you talk about "infinite objects" without defining them, then you aren't talking about mathematics. >How do you expect me to talk about infinity or infinite >sets without using the word "infinite"? If you can't, then you aren't talking about mathematics. -- Daryl McCullough Ithaca, NY
From: stephen on 20 Jul 2005 16:27 In sci.math Virgil <ITSnetNOTcom#virgil(a)comcast.com> wrote: > In article <MPG.1d48308522352190989f3d(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >> Inductive proof proves properties true for the entire set of naturals, right? > Wrong! It proves things only for the MEMBERS of that set, not the set > itself! > Definitions (Cantor): > (1) a set is finite if and only if there do not exist any > injective mappings from the set to any proper subset > (2) a set is infinite if and only if there exists any > injection from the set to any proper subset. > Clearly then, a set is finite if and only if it is not infinite. > Definitions (Auxiliary): > (3) a natural number, n, is finite if and only if the set > of naturals up to it, {m in N: m <= n}, is finite > (4) a natural number, n, is infinite if and only if the set > of naturals up to it, {m in N: m <= n}, is infinite > If these definitions are valid, then it is easy to prove buy induction > that there are no such things as infinite naturals: > (a) The first natural is finite, since there is clearly no > injection from a one member set the empty set. > (b) If any n in N is finite then n+1 is also finite. > This is also while quite clear, though a comprehensive proof > would involvev a lot of details. I can flesh it out a bit. Inductive step. Show that if n is finite, then n+1 is finite. Proof by contradiction. Suppose that n is finite, but that n+1 is infinite. This means there exists a bijection f from { 1, 2, 3, ... n+1} to some proper subset S of { 1, 2, 3, ... n+1}. Without loss of generality we can assume that S does not contain n+1. If we apply the function f to { 1, 2, 3, ... n} we get the set S-f(n+1). Because S does not contain n+1, S-f(n+1) is a proper subset of {1, 2, 3, ... n}. This means there exists a bijection from {1, 2, 3, .. n} to a proper subset of {1, 2, 3, ... n}, which means that n is infinite which contadicts the assumption that n was finite. Stephen
From: Tony Orlow on 20 Jul 2005 16:34
Virgil said: > In article <MPG.1d4816be96c6f78d989f36(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > Chris Menzel said: > > > > Aside from your tenuous grasp of basic transfinite arithmetic, the > > > critical terms in your argument -- "digital number", "length", > > > "exponentially longer" (cardinal or ordinal exponentiation?), "width", > > > "diagonal traversal", "below the line of diagonal traversal" -- are much > > > too vague for your argument to be evaluated. For all we know, you might > > > have some genuine insights. But currently, your argument is smoke and > > > mirrors; it hasn't been expressed as mathematics. > > > > I think you know exactly what i mean by each of those terms. > > In mathematics "think you know" is not good enough. When asked for > precise and comprehensive definitions in mathematics, it is the duty of > the user of any terms to give such precise and comprehensive definitions. > > > They are all > > widely understood. > > Not by me of by Chis. I can think of at least two antithetical meanings > for each of To's vague terms, so To piles ambiguity on ambiguity. > > > It is typical for Cantorians to resort to claims of vagueness on the > > part of their opponents, while presenting such vague proofs as the > > diagonal argument, without defining anything themselves, and assuming > > unfounded postulates such as all infinities are the same, except when > > they're not. > > Every bit of "Cantorianism" has been well enough defined for the > understanding of thousands upon thousands of people. That TO fails where > so many have succeeded says more about TO than about the adequacy of > "Cantorianism's" explanations. > Why don't you try asking for clarification of a term if you don't understand it? You are playing lawyer's interrogatory with me. I am not being sucked into that any more. Ask specific questions, insead of trying to pretend I am saying the "kerfloppen is infinitely gnudlio". -- Smiles, Tony |