[FOM] T(max) and Hilbert on physics

[Harvey Friedman]

I want to address something I said in my posting @05:Extremal Clauses,
1/18/04, 2:25PM.

I wrote that for 

T = theory of fields, T(max) is the theory of algebraically closed fields.

T = theory of Pythagorean planes, T(max) is first order complete Euclidean
geometry

T = theory of ordered fields, T(max) is theory of the ordered field of real
numbers (ordered real closed fields)

T = theory of real fields, T(max) is theory of the field of real numbers
(real closed fields).

The first two are fine, but the third makes no sense.

This is because the theory of real fields is not finitely axiomatizable,
where the first two theories are finitely axiomatizable.

Instead, I should have written:

T = theory of fields. T(max) + real field axioms is the theory of the field
of reals (real closed fields).

We can also proceed model theoretically. In particular, we can define T(max)
for any theory T to be the theory of the class of all models M of T such
that there is no proper M definable extension of M satisfies T.

Then we can meaningfully say

T = theory of real fields, T(max) is the class of all real closed fields.

On 1/23/04 11:34 AM, "Martin Davis" <martin at eipye.com> wrote:

> Harvey quotes Hilbert on this. In reading what Hilbert said, it is
> important to keep in mind that this was in 1900 before relativity and
> quantum theory, when it might appear that physics as a field for
> fundamental investigation was coming to an end, and what was left was to
> tidy up the loose ends.

Is there something that Hilbert wrote that you disagree with?

I doubt if there was ever a time where many reasonable people would agree
that "physics as a field for fundamental investigation was coming to an
end". 

Harvey Friedman