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

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.

> 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.

Martin

.

Relevant Pages

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