Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 27 Jul 2005 15:57 Chris Menzel said: > On Tue, 26 Jul 2005 16:44:24 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > > I am saying that if L CANNOT be infinite, then S^L CANNOT be > > infinite, > > No one disagrees with that, for fixed S and L. > > > and the fact that so many find this impossible to understand > > demonstrates that Poincare was right, and Cantorian transfinite > > cardinality is a disease in mathematics. > > Google for a recent post by Keith Ramsay for a correction of this > historical myth. > > > 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. See my inductive proof on the finiteness of N and respond to that. -- Smiles, Tony
From: Tony Orlow on 27 Jul 2005 15:58 malbrain(a)yahoo.com said: > Chris Menzel wrote: > > On Tue, 26 Jul 2005 16:44:24 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > > > I am saying that if L CANNOT be infinite, then S^L CANNOT be > > > infinite, > > > > No one disagrees with that, for fixed S and L. > > > (...) > > > > > 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. > > Obviously, he doesn't. Perhaps using the definition for all will help: > The set made by taking each and every (finite) L sequence of elements > of S is an infinite set. > > karl m > > So, the sum of a finite number of finite terms is infinite. Sure. -- Smiles, Tony
From: malbrain on 27 Jul 2005 16:00 aeo6 Tony Orlow wrote: > malbrain(a)yahoo.com said: > > Chris Menzel wrote: > > > On Tue, 26 Jul 2005 16:44:24 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > > > > I am saying that if L CANNOT be infinite, then S^L CANNOT be > > > > infinite, > > > > > > No one disagrees with that, for fixed S and L. > > > > > (...) > > > > > > > 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. > > > > Obviously, he doesn't. Perhaps using the definition for all will help: > > The set made by taking each and every (finite) L sequence of elements > > of S is an infinite set. > > > > karl m > > > > > So, the sum of a finite number of finite terms is infinite. Sure. No, there are an infinite number of elements in the infinite set. karl m
From: Robert Low on 27 Jul 2005 16:01 Tony Orlow (aeo6) wrote: > Robert Low said: >>OK, so how many elements are there in the set of all finite >>natural numbers? > 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. In fact, you can't even make sense.
From: malbrain on 27 Jul 2005 16:01
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. That's why I'm using the 1913 version of Webster's, before modern mathematics took sway over the definitions. karl m |