Obections to Cantor's Theory (Wikipedia article)
in [Logic]
|
Prev: Derivations
Next: Simple yet Profound Metatheorem
From: Virgil on 28 Jul 2005 00:33 In article <MPG.1d51c3927ce61a1b989fd3(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > There is no last set. The sequence goes on forever. > And when you have an infinite number of them, you still have finite > subscripts? How is that possible? By successively adding one to the previous value.
From: Virgil on 28 Jul 2005 00:39 In article <MPG.1d51c3c9a623009b989fd4(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > > >If you place no restriction on the length of strings, then they can be > > >infinitely long. > > > > No, there is only one restriction on the length of strings, and that > > is that the length is finite. > > > > -- > > Daryl McCullough > > Ithaca, NY > > > > > Oh, well you had said there was NO restiction, and they could get arbitrarily > large. I guess that wasn't true. It is true that finite strings can get arbitrarily large, meaning with more characters that any finite quantity. So that a set of finite strings with no finite limit on string length can be large enough to allow injections into proper subsets. This property is commonly abbreviated by saying that such a set is infinite (or Cantor-infinite for the prissy).
From: Virgil on 28 Jul 2005 00:45 In article <MPG.1d51c4219426ae86989fd5(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > 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. > That's false, no matter how many times you say it. There is no standard axiom system that requires, or even allows, its set of naturals to contain anything but finite members, though the set of all of them is necessarily Cantor-infinite. When, or if TO ever comes up with any such system of axioms of his own, we will be glad to point out its many self-contradictions and incompletenesses.
From: Virgil on 28 Jul 2005 00:55 In article <MPG.1d51c575971c9241989fd6(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > If I prove that there is no n in N for which the set of all strings > of length less than or equal to n is infinite, then I have proven > that no finite limit on the length of strings can produce an infinite > set. But you have not proven that in the absence of such a finite limit a set of strings cannot be infinite, and that is precisely the case. A set of finite strings but with no finite limit on length can be infinite. So that unless TO presumes a priori that a set of finite strings with no length limit, or a set of finite naturals with no size limit, is finite, he cannot prove it is finite. Circular arguments are not valid. And it is quite easy to show examples of such sets which allow injections into proper subsets, so that they are Cantor-infinite, even though they may be Orlow-finite.
From: Virgil on 28 Jul 2005 01:04
In article <MPG.1d51c6ff1e95e695989fd7(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > You folks are hopeless. We do not need your sick kind of hope when we have our healthy kind of logic and correctness. What is true in our axiom systems is true in them. What TO tries to argue in these systems contradicts their axioms and definitions all over the place, so he is wrong about what goes on and what can go on in these systems. TO has yet to produce any axiom system of his own, and, judging by the quality of his arguments, is extremely unlikely ever to be able produce one of any value. |