Re: what makes it true?

Torkel Franzen wrote:
> I was indicating a sense in which the concept of an ordinal, as used
> in the proof of Goldstein's theorem, can in fact be expressed in
> PA. Hence we must look elsewhere for an explanation of why the
> argument cannot be carried out in PA.

I don't think that expresses the concept of an ordinal, any more than
giving a list of cat names expresses the concept of a cat.

The proof of Goodstein's theorem requires all sorts of properties of
ordinals, including that every decreasing sequence of ordinals is
finite, definition of arithmetic operations on infinite ordinals as
well as finite ones, the use of well-ordering in proving that every
descreasing sequence is finite, that there is an ordinal greater than
every finite ordinal, ordering relations over their arithmetic, an
extension of the definition of Goodstein's sequence that preserves the
definition on finite ordinals, and so forth.

But expressing them is just the beginning. Then you have to somehow
prove (using only the deduction rules of PA) that every representation
of an operation or relation in finite ordinals corresponds to the
operation itself in natural numbers.


- Tim
.

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