# [FOM] Logic of irrational numbers

Timothy Y. Chow tchow at alum.mit.edu
Thu Mar 5 17:19:23 EST 2009

```Harvey Friedman <friedman at math.ohio-state.edu> wrote:
> For positive integers n and positive real numbers x, let x^[n] =
> x^...^x, where there are n >= 1 x's.
>
> WILD CONJECTURE 1. There exists n >= 1 such that "e^[n] is irrational"
> can be proved using large cardinals, but not in ZFC.

It follows from Schanuel's conjecture that e^[n] is transcendental for
all n.  As I mentioned before, Schanuel's conjecture says that if a_1,
a_2, ..., a_k are linearly independent over the rationals, then the
transcendence degree of Q[a_1, a_2, ..., a_k, exp(a_1), exp(a_2), ...,
exp(a_k)] over Q is at least k.  Taking a_i = e^[i] yields the claimed
result.

So if we let "Schanuel[k]" be Schanuel's conjecture restricted to a
particular value of k, then the "WILD CONJECTURE" might be more naturally
(though perhaps less strikingly) replaced by:

There exists k >= 1 such that Schanuel[k] is provable using
large cardinals but not in ZFC.

Tim
```