More powerful axioms [for mathematics]
If one looks for example at progressively more complicated Diophantine equations then one can expect that one will find examples where more and more powerful axiom systems are needed to prove statements about them. But my guess is that almost as soon as one reaches cases that cannot be handled by Peano arithmetic one will also reach cases that cannot be handled by set theory or even by still more powerful axiom systems.
Any statement that one can show is independent of the Peano axioms and at least not inconsistent with them one can potentially consider adding as a new axiom. Presumably it is best to add axioms that allow the widest range of new statements to be proved. But I strongly suspect that the set of statements that cannot be proved is somehow sufficiently fragmented that adding a few new axioms will actually make very little difference.
In set theory (see page 1155) a whole sequence of new axioms have historically been added to allow particular kinds of statements to be proved. And for several decades additional so-called large cardinal axioms have been discussed, that in effect state that sets exist larger than any that can be reached with the current axioms of set theory. (As discussed on page 816 any axiom system that is universal must in principle be able to prove any statement that can be proved in any axiom system—but not with the kinds of encodings normally considered in mathematical logic.)
It is notable, however, that if one looks at classic theorems in mathematics many can actually be derived from remarkably weak axioms. And indeed the minimal axioms needed to obtain most of mathematics as it is now practiced are probably much weaker than those on pages 773 and 774.
(If one considers for example theorems about computational issues such as whether Turing machines halt, then it becomes inevitable that to cover more Turing machines one needs more axioms—and to cover all possible machines one needs an infinite set of axioms, that cannot even be generated by any finite set of rules.)
