Obections to Cantor's Theory (Wikipedia article)
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From: Daryl McCullough on 27 Jul 2005 16:00 Tony Orlow says... >How silly will you feel when you finally come to realize that this unending >stream of "cranks" were all right, while you maligned, isnulted and bullied >them? Will you ever apologize, or go to your graves in denial? Gee, I >wonder.... I'm sure that Torkel Franzen will apologize to you after you are finished generating all finite natural numbers. -- Daryl McCullough Ithaca, NY
From: David Kastrup on 27 Jul 2005 16:21 malbrain(a)yahoo.com writes: > David Kastrup wrote: >> malbrain(a)yahoo.com writes: >> >> > David Kastrup wrote: >> >> malbrain(a)yahoo.com writes: >> >> >> >> > malbr...(a)yahoo.com wrote: >> >> >> Tony Orlow (aeo6) wrote: >> >> >> > Some finite, indeterminate number. You tell me the largest >> >> >> > finite number, and that's the set size. It doesn't exist? >> >> >> > Well, then, I can't help you. >> >> >> >> >> >> >From websters 1913 dictionary: >> >> >> >> >> >> De*ter"mi*nate (?), a. [L. determinatus, p. p. of determinare. See >> >> >> Determine.] >> >> >> >> >> >> 1. Having defined limits; not uncertain or arbitrary; fixed; >> >> >> established; definite. >> >> >> >> >> >> >> >> >> Thus "indeterminate" is the exact opposite of fine. You can't have it >> >> >> both ways. >> >> > >> >> > Ooops. "indeterminate" is the exact opposite of finite. You've >> >> > uncovered a contradiction about the count of elements in an >> >> > infinite set that cannot be resolved from the definitions of >> >> > finite and indeterminate. karl m >> >> >> >> Uh, no. "Indeterminate" just means unspecified, not infinite. >> > >> > Read the definition. Determinate=defined limit; >> > indeterminate=undefined limit=infinite. >> >> Undefined limit is not the same as infinite. For example, in >> programming languages indeterminate loop forms are those for which you >> can't say in advance how often they will be run (i.e., while-loops, as >> opposed to the determinate for-loops). > > Programming languages weren't invented in 1913. Please use > analogies that are pre-Cantor. The opposite of "undefined-limit" is > infinite. Algorithms are definitely pre-Cantor. Euklid's algorithm (a few thousand years old) for finding the greatest common divisor of two numbers takes an indeterminate number of iterations. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Tony Orlow on 27 Jul 2005 16:27 Han de Bruijn said: > Tony Orlow (aeo6) wrote: > > Han de Bruijn said: > >>Set theory doesn't deserve such a predominant place in mathematics. > >>After the discovery of Russell's paradox et all, everybody should have > >>become most reluctant. > > > > Not to mention the Banach-Tarski spheres. Doesn't that derivation constitute a > > disproof by contradiction? Isn't the result absolutely nonsensical? And yet, it > > is accepted, somehow, as truth that one can chop a ball into a finite number of > > pieces and reassemble them into two solid balls, each the same size as the > > original, despite all evidence and logic to the contrary. It's a clear sign of > > something wrong in the system, when it produces results like that. > > Not to mention Goodstein's theorem, which states that some (ugly formed) > sequences always converge. They prove this finitary statement with help > of infinite ordinal numbers. If the last few axioms of ZFC are rejected, > the finitary result still does exist, while the infinitary "proof" just > vanishes into nothingness. How is that possible? > > Han de Bruijn > > I just looked up Goodstein's Theorem, and it looks interesting. I'll have to make a note to check this one out at some point. Right now, I don't have any answer for you. The construction of his numbers is interestingly similar to my base-free enumeration of the real number circle, though, so it may be pertinent to some other things I am thinking about. Thanks for the pointer! -- Smiles, Tony
From: Daryl McCullough on 27 Jul 2005 16:09 Tony Orlow <aeo6(a)cornell.edu> said: >Chris Menzel said: >> > For finite S, S^L can ONLY be infinite with infinite L. Why is this so >> > hard to understand? If S and L are both finite, then S^L is finite, >> > isn't it? >> >> Yes, of course. But that's for a fixed L, say 17. But for any given >> nonempty set S of natural numbers, the set of *all* finite sequences of >> elements of S -- i.e., S^1 U S^2 U S^3 ... -- is infinite. That's the >> part you don't seem to get. > >Not unless you allow L to go to infinity. You keep bringing up L. There *is* no L. There *is* no maximum length for finite strings. It's not that the maximum length is infinite, it's that there *is* no maximum length. -- Daryl McCullough Ithaca, NY
From: Virgil on 27 Jul 2005 16:27
In article <MPG.1d517f695ae1a2f3989fa6(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > If you place no restriction on the length of strings, then they can be > infinitely long. If you place only the restriction that no string be infinite, the set of strings can still contain strings longer that any finite length. If they cannot be infinitely long, then they are restricted > to > finite lengths, and the set size is restricted to a finite size. What is that alleged finite set size? And why can it not be made greater by 1? Does TO claim that any set of finite strings cannot be made to hold one more string? Unless he claims that, there is no finite limit on the size of such a set. > Set theorists need to get back to using real numbers more often. Reals are for analysts, not set theoreticians. > > > > >If S is finite (2 for binary, etc) then we must allow infinite > > >strings, in order to have infinite numbers of strings. Only TO is so limited. Normal people can get infinite sets of finite strings just like they can get infintie sets of finite naturals. TO really cannot impose his own shortcomings on others, however hard he tries. > > > > Why do you keep saying that? It's provably false. The set of all > > finite strings is an infinite set. It's infinite by *your* definition > > of infinite, in the sense that it is "without end". The set of all > > finite strings is the union of > > > > S_1 = the set of strings of length 1 > > S_2 = the set of strings of length 2 > > S_3 = the set of strings of length 3 > > ... > > > > The collection of subsets S_n goes on without end. > So, each of these sets is finite right, given finite S and L? > There are an infinite number of such finite sets? > Do they then go, say, from S_1 to S_oo? > And S_1 is the set of strings of length 1, and S_2 is the set of strings of > length 2, etc, so S_n is the set of strings of length n? > Okay. What length are the strings in S_oo? There is no S_oo, only S = Union_{n finite and in N} S_n, and it contain strings of all finite natural number lengths, but with no such thing as a longest finite string, as there is always a longer fintie string. To deny this, as TO tries to do, requres that there exist a finite string so long that one cannot append another character to its end. |