[FOM] Clarification on Higher Set Theory

[sean.stidd@juno.com]

One of the topics of recurring interest on this list (and in the FOM community more generally) is the potential for 'justifying belief in' (despite the problems with this idiom I know no better) higher set theory by way of potential 'applications' to core mathematical theories outside set theory and, perhaps, to the mathematical theories central to physics.

I take it that the goal of such research would be finding proofs of statements in such a mathematical or physical theory supplemented with higher set-theoretic assumptions that are

(a) provably not provable or refutable in the (non-supplemented) mathematical or physical theory in question, and

(b) have a clear sense when formulated in that theory, and which perhaps 'strike one as true' (in the sense that the (extensionally formulated) axiom of choice does this) given the concepts of that theory; or, alternatively and perhaps better, which have clear and important consequences for 'going problems' in the rest of that theory.

My understanding from the list is that such results are in fact emerging from the work of Professor Friedman and others, but it would be nice to have some specific examples to chew on: a proposition formulable within intuitive number theory, geometry, group theory, etc. which one could present to the moderately well mathematically educated and say: "this is why you should believe in higher set theory."