Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 27 Jul 2005 13:28 malbrain(a)yahoo.com said: > Tony Orlow (aeo6) wrote: > > malbrain(a)yahoo.com said: > > > Virgil wrote: > > > > In article <MPG.1d4863d52071fde5989f51(a)newsstand.cit.cornell.edu>, > > > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > > > > 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 > > > > > > Under Tony's theory, the number representing N is defined by a string > > > of an infinite number of ones. Yes, more than one Turing machine can > > > produce this. karl m > > > > > > > > Actually there are two ways to look at it. In unsigned binary, yes, an infinite > > number of 1's is the largest number possible. Since we start with all 0's > > representing 0, the size of the set, N, will be one more than 111...111. It > > will be 000...001:000...000, or one unit infinity. > > There's already a STANDARD method for coding real numbers using the > integers. It's published by the Institute of Electrical and Electronic > Engineering. Those aren't real numbers. Their infinite whole numbers. > > > Now, the other way to look at things, from a computer science standpoint, is > > with 2's complement, the signed binary integer representation used in just > > about every computer on the planet. A string of all zeroes, or 0, is obviously > > it's own negative. A string of all 1's, rather than being the largest number, > > is -1. The largest positive number is 011...111, and the largest negative is > > generally considered to be 100...000. BUT, this number is its own negative > > value. It is both positive and negative, being at once the largest positive and > > negative. 100...000 is really infinity, in the scheme of the signed binary > > number circle, and just like + and -0 are equivalent, in this perspective + and > > -oo are also equivalent. > > You still haven't answered Daryl's questions from yesterday evening. I've been catching up after a long weekend away. > karl m > > -- Smiles, Tony
From: malbrain on 27 Jul 2005 13:27 Robert Low wrote: > Tony Orlow (aeo6) wrote: > > Proof that f(n), the number of strings in the set of all strings up to and > > including length n in N, on a finite alphabet of size S, is finite: > > But nobody has disagreed with that. The point of contention is > whether the union over all n of S^n contains a finite string; > it doesn't, but you claim it does. I think you meant whether the union contains an INFINITE string. karl m
From: Tony Orlow on 27 Jul 2005 13:30 malbrain(a)yahoo.com said: > David Kastrup wrote: > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > > > Virgil said: > > >> In article <MPG.1d489d7fea8af732989f60(a)newsstand.cit.cornell.edu>, > > >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > >> > > >> > I have explained the flaw in this proof, and it is met with > > >> > confusion because none of you seems to appreciate the recursive > > >> > nature of inductive proof. > > >> > > >> The inductive axiom shortcuts that recursion, which is the point of > > >> the inductive axiom. It says that if the recursive step can be > > >> proved in general, then it never need be applied recursively. > > >> > > >> If TO wishes to reject the inductive axiom, only then can he argue > > >> recursion. > > > > > > What a load of bilge water! Accepting the axiom as a general rule > > > does not mean one has to immediately forget about the lgical basis > > > for the axiom. > > > > Of course not. But the basis for choosing an axiom is irrelevant to > > the application of the axiom. And this axiom was chosen exactly in a > > manner that does not require recursive application. > > When working a proof one chooses one's axioms based on the desired > result. How can you say that the choice is irrelevant to the > application? > > > Whether it was chosen because it is equivalent to arbitrarily deeply > > nested recursion or because Kronecker's neighbor had a particularly > > ugly elephant locked in his belfry is irrelevant to its application. > > Do you know the actual history that went into its development as an > axiom? thanks, karl m > > I'd be interesting in hearing about that, Karl. -- Smiles, Tony
From: Tony Orlow on 27 Jul 2005 13:31 malbrain(a)yahoo.com said: > Dik T. Winter wrote: > > In article <1122347583.518181.245300(a)g14g2000cwa.googlegroups.com> malbrain(a)yahoo.com writes: > > > > > > The C language is defined by the C standard, as defined by ISO. There > > > are no "unbounded" standard types in the C language. karl m > > > > Who is talking about C? > > Of the billions of computer systems deployed since the micro-computer > revolution, the overwhelming majority are programmed with C. karl m > > C IS the best language. I love C. -- Smiles, Tony
From: Daryl McCullough on 27 Jul 2005 13:10
Tony Orlow says... >despite the fact that an infinite set of whole numbers requires >infinite whole numbers That's false, no matter how many times you say it. No finite set can contain every (finite) natural. Why? Because every finite set of naturals has a largest element, and there is no largest finite natural. -- Daryl McCullough Ithaca, NY |