When it comes to Type 1 evidence (from the practice of set theory as mathematics) we don’t require that opinions about what is “good set theory” be shared (and “the picture” is indeed determined by “good set theory”). As Peter put it:
” Different people have different views of what “good set theory” amounts to. There’s little intersubjective agreement. In my view, such a vague notion has no place in a foundational enterprise.”
What Peter wrote is this:
The notion of “good set theory” is too vague to do much work here. Different people have different views of what “good set theory” amounts to. There’s little intersubjective agreement. In my view, such a vague notion has no place in a foundational enterprise. The key notion is evidence, evidence of a form that people can agree on.
I probably should have stepped in at the time to remark that I’ve been using the term ‘good set theory’ for the set theory that enjoys the sort of evidence Peter is referring to here — for example, there was evidence for the existence of sets in the successes of Cantor and Dedekind, and more recently for PD, in the results cited by Peter, Hugh, John Steel, and others. (Using the term ‘good set theory’ allows me to leave open the question of Thin Realism vs. Arealism. For the Arealist, these same considerations are just reasons to add sets or PD to our mathematics/set theory, but the Thin Realist sees them as evidence for existence and truth.)
This doesn’t preclude people disagreeing about what parts of set theory they believe to be more interesting, important, promising, etc. (Scientists also disagree on such matters.)
At the present juncture, it’s more difficult to find and assess new evidence, but that’s to be expected. Peter and Hugh have made it clear, I think, that they regard many of the current options as open (including HP, when it begins to generate definite claims), that more information is needed. If one theory eventually ‘swamps’ the rest (I should have noted that ‘swamping’ often involves ‘subsuming’), then the apparently contrary evidence will have to be faced and explained. (Einstein had to explain why there was so much evidence in support of Newton.)