FOM: Goedel Theorem

[William Tait]

Dear Mr Drago,

Since Goedel's incompleteness theorems do not mention finitism, I'm
assuming that your question

> What occurs when in Goedel theorem one
>puts as the "finite" mathematics Hilbert allowed in metamathematics Bishop's
>mathematics, which is well-defined both in mathematical terms (no more than
>potential infinity) and in logical terms (non-classical logic)?

concerns whether the consistency of Peano arithmetic can be proved in
Bishop's theory.

It is not entirely clear that Bishop's constructive mathematics *is*
well-defined, but he did, in *Foundations of Constructive Analysis*, admit
the inductive definition of the Borel sets. This kind of inductive
definition is far more than enough to prove induction up to \epsilon_0 and
so prove the consistency of Peano Arithmetic.

>common reference about "finite"
>mathematics is Tait "Finitism", J. Phil. 1981 which moreover is in my opinion
>unsatisfactory in identifying "finitism" with primitive recursive functions.

There are really two questions: one is the conceptual question of making
sense of the `finite' part of `finitism', when one is contemplating
finitist proofs of polynomial equations with free variables and so,
apparently, referring to the infinite totality of numbers. (Your reference
to the distinction between `potential' and `actual' perhaps comforts you
more than it does me. But in any case, as you yourself use it, it does not
distinguish finitism from intuitionism or Bishop's constructive
mathematics.)  I tried to solve this problem by giving a sense to reasoning
about an *arbitrary* number, as opposed to reasoning about *all* numbers. I
confess to being still satisfied with that analysis. I note that Goedel
also, in the unpublished *1933o and *1938a, seems to have identified
finitism with PRA.

The other question is the historical one of what *Hilbert* meant by
`finitism'. Many people have questioned the identification of finitism with
PRA, without making clear whether they are questioning the conceptual
thesis or the historical one.

I, for one, would be very interested in a further analysis of either
question. But it won't do simply to say that you don't agree with *the*
thesis.

Regards,

Bill Tait