[FOM] logical complexity of Schanuel's Conjecture

[Dave Marker]

Lou van den Dries has noticed that Macintyre's result can be strengthened
so that
if there is a counterexample to Schanuel's conjecture there is a primitive
recursive
counterexample.

Using this you can eliminate the use of the Low Basis Theorem in the
argument I posted
yesterday and just use a recursive enumeration of the primitive recursive
complex numbers.

Dave

David Marker
LAS Distinguished Professor
Mathematics, Statistics, and Computer Science
University of Illinois at Chicago

On Fri, Oct 23, 2015 at 11:48 AM, Dave Marker <marker at math.uic.edu> wrote:

> Joe Shipman asked if Schanuel's Conjecture is equivalent to an
> arithmetic statement.
>
> Angus Macintyre in his preprint "Turing meets Schanuel" shows, among
> other things,
> that if there is a counterexample to Schanuel's Conjecture than there
> is a counterexample with among the recursively coded complex numbers.
> (This preprint will be part of the proceedings of the Logic Colloquium
> held a couple of years ago
> in Manchester)
>
> This gives, I think, a Pi^0_3-statement equivalent to Schanuel's
> conjecture.
>
> I think that with a couple of tricks you can improve this to obtain an
> equivalent
> Pi^0_2 statement.
>
> I've posted a link to a sketch of this
>       http://www.math.uic.edu/~marker/SC2.pdf
>
> Dave
>
> David Marker
> LAS Distinguished Professor
> Mathematics, Statistics, and Computer Science
> University of Illinois at Chicago
>
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