Dear Sy,

Type 2 comes down HARD for Forcing Axioms and V = L, as so far none of the others has done anything important for mathematics outside of set theory.

I was assuming that any theory capable of ‘swamping’ all others would ‘subsume’ the (Type 1 and Type 2) virtues of the others. It has been argued that a theory with large cardinals can subsume the virtues of V=L by regarding them as virtues of working within L. I can’t speak to forcing axioms, but I think Hugh said something about this at some point in this long discussion.

All best,

Pen