Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 25 Jul 2005 15:41 In article <MPG.1d4eceec82a09325989f6c(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > David Kastrup said: > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > > > And yet, one cannot apply an increment an infinite number of time > > > without adding infinity. > > > > It is neither necessary nor feasible to increment "an infinite number > > of time" or "add infinity". > The Peano axioms do just that. Those axioms only do it once per number. One of the trivial consequences of those axiom is the theorem that any non-empty set of naturals has a first (smallest) member. If TO were right that the set of infinite naturals is non-empty, there would have to be a smallest one, anddit would have to have a predecessor which is finite, unless TO wants to say that all naturals, including 1, are infinite. So that either there are no infintie naturals or all naturals are infinite.
From: MoeBlee on 25 Jul 2005 15:46 >From a post by Tony Orlow (aeo6): > Why do you think inductive proof is agreed to work? It doesn't have to be just agreed to work. It is proven to work. It is proven by the fact that inductive proofs are applied to inductive sets. The very nature of a set being inductive is what ensures inductive provability. > Think. You think. And read a book about the subject, would you please?
From: Tony Orlow on 25 Jul 2005 15:52 Martin Shobe said: > On Wed, 20 Jul 2005 10:57:58 -0400, Tony Orlow (aeo6) > <aeo6(a)cornell.edu> wrote: > > >Barb Knox said: > >> In article <MPG.1d4726e11766660c989f2f(a)newsstand.cit.cornell.edu>, > >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > >> [snip] > >> > >> >Infinite whole numbers are required for an infinite set of whole numbers. > >> > >> Good grief -- shake the anti-Cantorian tree a little and out drops a > >> Phillite. Here's a clue: ALL whole numbers are finite. Here's a > >> (2nd-order) proof outline, using mathematical induction (which I > >> assume/hope you accept): > >> 0 is finite. > >> If k is finite then k+1 is finite. > >> Therefore all natural numbers are finite. > >> > >> > >That's the standard inductive proof that is always used, and in fact, the ONLY > >proof I have ever seen of this "fact". Is there any other? I have three proofs > >that contradict this one. Do you have any others that support it? > > > >Inductive proof proves properties true for the entire set of naturals, right? > > Yep. > > >That entire set is infinite right? > > Yep. > > >Therfore, the number of times you are adding > >1 and saying, "yep, still finite", is infinite, right? > > Yep. But be careful here, at *every* stage of this process, we have > still only done it a finite number of times. Uhhh.... Look at what you just agreed to. The number of times you are adding 1 is infinite. But, now you say it is always a finite number of times? make up your mind. > > > So, you have some way of > >adding an infinite number of 1's and getting a finite result? > > Nope. You weren't careful. You contradicted yourself. Do you apply successor and increment the value a finite number of times, or an infinite number of times? Be careful. > > > You might want to > >discuss this with your colleagues specializing in infinite series. There is a > >very simple rules that says no infinite series can converge to a finite value > >unless the terms of the series have a limit of zero as n approaches infinity. > >Does this constant term, 1, have a limit of zero? > > Nope. > > > No it doesn't, and the > >infinite series of constant 1's cannot converge, but diverges to infinity. > > Yep. > > > Can > >you actually deny this? If so, then Poincare was right. > > BTW, there is a caveat on convergence. You have to assume the > standard topology. In other topologies, that sequence can converge. You mean with a ring? That's really not what we're talking about, unless you agree that the number line is a circle, and even then it's not relevant. In pure quantitative terms, a sum of infinite 1's is infinite. So, please make up your mind. Do we increment to get a successor an infinite number of times, or only a finite number of times, to get N? > > Martin > > -- Smiles, Tony
From: Tony Orlow on 25 Jul 2005 15:52 Virgil said: > In article <MPG.1d4858812235c0b0989f4c(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > David Kastrup said: > > > > What is supposed to be a "constant equality"? > > > > An equality that holds true for n=1, and for n=n+1 given true for n. > > In mathematics, n=n+1 is always false. > excuse the typo. again -- Smiles, Tony
From: david petry on 25 Jul 2005 15:53
Jiri Lebl wrote: > > Mathematicians were solving real world problems long before Cantor's > > Theory came along. > > Set theory made it possible to get much more profound > mathematical results that were then applied to real systems where such > predictions agreed with observation. I doubt you could come up with an example that survives close scrutiny. (i.e. where it was undeniably set theory that made the result possible) > > Applied mathematicians never need to think about > > completeness, and I suspect many don't know what it is. > > That's the whole point of completeness, that you can just take limits > willy nilly and assume you get a real number back as long as the > sequence is cauchy So it helps applied mathematicians sleep better at night? Funny how mathematicians before Cantor would never have had any doubts about taking limits. > > My argument is that the mathematics that is relevant to real > > world problems necessarily has a property of "observability" - that is, > > mathematical statements must have observable implications, where > > we think of the computer as the mathematician's microscope which > > lets us make observations. > What is this obsession with "computer as a microscope"? Computer is a > very limitted device capable of making some limited formal > manipulations on strings of meaningless characters. I'm talking about computers in the abstract, as a conceptual aid to clarify what mathematics is all about. |