Re: Obections to Cantor's Theory (Wikipedia article)

Martin Shobe wrote:
> On 26 Jul 2005 22:07:39 -0700, malbrain@xxxxxxxxx wrote:
>
> >Martin Shobe wrote:
> >> On 26 Jul 2005 17:15:10 -0700, malbrain@xxxxxxxxx wrote:
> >>
> >> >Chris Menzel wrote:
> >> >> On Tue, 26 Jul 2005 16:39:58 -0400, Tony Orlow <aeo6@xxxxxxxxxxx> said:
> >> >> > ...
> >> >> > then that function needs to be taken into account. This nonsense
> >> >> > about an infinite set of finite whole numbers is pretty bad too, but
> >> >> > probably without any real consequences.
> >> >>
> >> >> You seem to agree that the set of whole numbers is infinite. But there
> >> >> was an inductive argument a few posts back that all the whole numbers
> >> >> are finite, and hence that the set of finite whole numbers is infinite.
> >> >> There was some real mathematics there.
> >> >
> >> >How does it follow that the count of finite whole numbers is infinite?
> >> >How is this established by the Peano axioms?
> >>
> >> A set, A, is infinite if, and only if, there exists a one-to-one
> >> function, f:A -> A, such that f(A) is a proper subset of A.
> >>
> >> Or equivalently,
> >>
> >> A set, A, is infinite if, and only if, there exists a function, f:A ->
> >> A, such that
> >> 1) for all x,y in A, f(y)=f(x) => x=y.
> >> 2) there exists an x in A such that for all y in A, f(y) =/= x.
> >>
> >> Since there are no sets, and we are interested only in the domain of
> >> PA, we have
> >>
> >> The domain of PA is infinite if, and only if, there exists a function,
> >> f, such that
> >> 1) for all x,y f(y)=f(x) => x=y.
> >> 2) there exists an x such that for all y, f(y) =/= x.
> >>
> >> The successor function meets those criteria. Therefore, the domain of
> >> PA is infinite.
> >>
> >> The only problem that I can see with this is that it's a theorem about
> >> PA instead of a theorem of PA.
> >
> >It's ABOUT PA because you back-filled a definition for infinite to PA?
>
> It's about PA because the question isn't even expressable in PA.
> There are no functionn in the domain of PA, just numbers.
>
> >> > You have Tony agreeing to
> >> >the axiom of infinity apriori, when this is not indicated.
> >>
> >> The axiom of infinity is not needed to prove that a set is infinite.
> >> The axiom of infinity is needed to prove that infinite sets exist.
> >
> >This KOAN is going to be a tough sell. karl m
>
> What do you think the axiom of infinity says? I'm not looking for the
> logical consequences, just what the axiom is.

Well, perhaps I need to define KOAN first before I answer your
question.

Etymology: Japanese kOan, from kO public + an proposition
: a paradox to be meditated upon that is used to train Zen Buddhist
monks to abandon ultimate dependence on reason and to force them into
gaining sudden intuitive enlightenment.

The axiom of infinity imports the set of natural numbers into set
theory.

karl m

.

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