[FOM] Primitive recursive reals

[Timothy Y. Chow]

Harvey Friedman <friedman at math.ohio-state.edu> wrote:
> QUESTION: Is e + pi primitive recursive in base 2, base 10, or even (as 
> expected) primitive recursive in every base?

This looks difficult.  It isn't known whether e+pi is irrational.  Of 
course, one might still be able to answer the above question affirmatively 
along the lines of, "If e+pi is rational then it is primitive recursive in 
every base; if e+pi is irrational then...."  However, given that we don't 
know how to control the diophantine approximation properties of e+pi well 
enough to prove its irrationality, the bare assumption that e+pi is in 
fact irrational is unlikely to give enough of an extra boost to get the 
primitive recursive bound that it seems one needs to answer Harvey 
Friedman's question.

Tim