Axiomatics have been problematic since Godel, Tarski, Skolem, and Church,
and, influenced by Brouwer and Wittgenstein I think, various forms of
conventionalism are becoming dominant. But this is happening at the same
time that physics is heading towards mathematical platonism in relation
to the real - so who knows what will emerge?
Also, infinitesimals have to be taken into account here; even the con-
tinuum is under revision.
For myself, I tends towards a Platonism, much as Godel did, although a
friend of mine, working with the Russian mathematician Alexander
Essenin-Volpin, tends towards what he calls ultra-intuitionism, a radical
form of intuitionism which argues that perception and construction, not
closure, are at the heart of mathematics.
Alan, hoping he's not intruding