Set theory:

See http://plato.stanford.edu/entries/quine-nf/ on an alternative to ZF …

and lots else all over the web on alternatives to the axioms of ZF.

Fractals:

- Any relationship between the maths of fractals and spacetime is good (i.e. is philosophical).

- Are coastlines etc. really fractal even though they obviously don't have the structure of any fractal set described in pure mathematics?

- Orderings on fractal sets.

- Fractals and computability (computation of fractal sets by machines more or less powerful than Turing machines). Relationship between ordering and computability.

- Something about the axiom of choice. (But maybe fractal stuff only needs the Axiom of Finite Choice, which is much easier to get in all the set theories we know.)

Combinations:

- Can we describe fractals over alternatives to ZF?