# [FOM] A question on Natural Arithmetical Independence

Harvey Friedman friedman at math.ohio-state.edu
Sat Nov 15 09:15:48 EST 2003

On 11/14/03 8:53 PM, "Dmytro Taranovsky" <dmytro at mit.edu> wrote:

> Consider the following statement:
> The sum of the binary digits of the number of functions (from sequences
> of 100 integers to integers) that are polynomial in 100 variables with
> degree less than 1000 (note: x*y is a polynomial of degree 2) and
> coefficients that are integers of absolute value less than 2^1000, and
> have no zeroes   is even.
>
> Is the statement independent of Peano Arithmetic?  If so, how do you
> prove it?  Is it independent of ZFC+(there exists a supercompact
> cardinal)?  If so, how do you prove the independence under the
> assumption that there is a countable transitive model of ZFC+(there
> exists a supercompact cardinal)?
>
> If the statement is decidable in ZFC, would a simple modification (which
> one?) make it undecidable?
>
>

I have known about such attempts for decades, but I never saw how to control
the situation enough to get parity (even/odd) to do the work.

On the other hand, I suspect something like this can be pushed through.

Recall my previous FOM posting about things like

sin(2^2^2^2^2^2^2^2^2^2) > 0.

Here the (totally unresolved) issue is lengths of proofs.

Harvey Friedman