# [FOM] irrational conjectures

Harvey Friedman friedman at math.ohio-state.edu
Sat Mar 7 15:01:58 EST 2009

```As pointed out by Joe Shipman and Tim Chow, the statement

For all n, e^[n] is irrational

follows from Schanuel's Conjecture, and so making the wild conjecture
that for some n, the statement is provable from large cardinals and
not from ZFC, implies the statement that Schanuel's Conjecture is not
provable from ZFC. Of course, it also carries the suggestion that
Schanuel's Conjecture might be provable from large cardinals.

Here is another statement about n:

sin(2^[n]) > 0.

where 2^[n] = 2^2^...^2, with n 2's.

Since sin(n) is nonzero for all integers n >= 1, the above statement,
for any n, is provable or refutable in extremely weak fragments of PA.

WILD CONJECTURE. There exists a positive integer n < 2^1000 such that
the statement sin(2^[n]) > 0 can be proved using (commonly studied)
large cardinals using at most 2^20 symbols, but cannot be proved in
ZFC using at most 2^2^2^20 symbols.

Harvey Friedman

```