FOM: 65:Simplicity of Axioms/Conjectures

[Matthew Frank]

In his post #65, Harvey Friedman conjectured that

CONJECTURE 1.1. Let phi be a formula in the language of PA with at most 20
symbols. The universal closure of phi is provable or refutable in PA.

I just wanted to point out that this conjecture, even if true as stated,
would be very hard to extend at all past the number 20.  For instance, (so
far as I can tell from a MathSciNet search) it is still an open question
in number theory whether there are infinitely many primes of the form x^2
+ 1, i.e. whether the universal closure of the 23-symbol formula

(exists b)(forall c)(forall d) not s[(a+b)*(a+b)] = ssc * ssd

is true.  (The best result seems to be by Iwaniec in 1978, that infinitely
many numbers of form x^2 + 1 are products of two primes.)

Unfortunately, I haven't figured out the relationship of Conjecture 1.1
to the other conjectures in Friedman's post #65, and so don't know how
relevant this is to the ideas that the axioms of number theory are forced
on us by their simplicity.  The only moral I can draw is that, in a list
of formulas by simplicity, open problems will show up much before the 
formulas we can currently prove undecidable.

--Matt Frank

(since I am new to the list:
Name:  Matthew Frank
Position:  graduate student in Conceptual Foundations of Science
Institution:  University of Chicago
Research Interests:  history of math, foundations of math
More info:  http://www.math.uchicago.edu/~mfrank)