Angus J. MacIntyre:
How much has mathematics been affected by Gödel's work?
Gödel's dramatic work seems to have affected mathematics (beyond logic)
very little. In particular, it fits badly with the geometrical ideas
widespread in contemporary mathematics. Of course, in set theory his
influence has been profound, and thanks to him and Cohen, and many very
gifted followers, one knows wonderful things about models of ZF. But
this has not led to organic connections to other parts of mathematics, and
indeed set theory is remote from the centre of mathematics.
In number theory and geometry, no one has ever detected a hint of the
Incompleteness Phenomena around problems perceived by practitioners as
central or natural. What we now know about unprovability of consistency,
undecidability, or the effect of large cardinals on the unsolvability of
diophantine equations,has neither induced paralysis or anxiety in
mathematicians, nor a rush to understand the fine detail of large
Technical ideas of Gödel (outside set theory) remain useful, for
example around functional interpretations (relevant to the unwinding of
proofs in the area where number theory meets ergodic theory and hard
analysis). Refinements of his use of the Chinese Remainder Theorem were
central to the negative solution of Hilbert's 10th problem (though quite
new ideas, and probably geometric ones, appear to be needed for the
problem for the rationals). Even in the integer case, the result is about
situations far removed from foreseeable concerns of number theorists.
Again, in combinatorial group theory one saw first a wave of negative results
(a la Gödel) around decidability, then a brilliant twist by Higman
to get hold of subgroups of finitely presented groups. But now, these
ideas fade, and the geometrical ideas of Gromov dominate.
It is to be noted that large parts of mathematics are provably
nonGödelian. Notable examples are the "o-minimal" universes, relating to
a slogan of Grothendieck about "topologie moderee", where one has enough
expressive power to do geometrical things freely, but not so much that
one gets dragged into the geometrically irrelevant
pathologies of set-theoretic analysis. Tarski's work on real-closed
fields provided the first example, and now many other richer ones are known.
Gödel's work seems irrelevant here.