[FOM] A question on Natural Arithmetical Independence
Harvey Friedman
friedman at math.ohio-state.edu
Sat Nov 15 09:15:48 EST 2003
Reply to Taranovsky.
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
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