Well, obviously we could simply take every true sentence of Peano arithmetic as an axiom to obtain a consistent and complete system, but if we think in that spirit, then almost every mathematician in the world is working on finding a better set of axioms (because every proof would either give us new axiom or show that something should not be included as axiom), right?
> obviously we could simply take every true sentence of Peano arithmetic as an axiom to obtain a consistent and complete system
If you’re talking about every true sentence in the language of PA, then not all such sentences are derivable via the theory of PA. If you are talking about the theorems of PA, then these are missing an infinite number of true statements in the language of PA.
Harvey Friedman’s “grand conjecture” is that virtually every theorem that working mathematicians actually publish can already be proved in Elementary Function Arithmetic (much weaker than PA in fact). So the majority of mathematicians are not pushing the boundaries of the existing foundational theories of mathematics, although there is certainly plenty of activity regardless.
