I think it is great that you are getting interested in the philosophy
and foundations of set theory but you really have to do your homework
- You take it as a given that the principles of Zermelo set theory
follow from (that is, are intrinsically justified on the basis of) the
iterative conception of set. This is highly contested. See, for
example, the classic paper of Charles Parsons on the topic.
- You say that when Pen’s “Thin Realist talks about some statement being true as a result of its role for producing “good mathematics” she almost surely means just “good Set Theory” and nothing more than that.”
I think you have misunderstood Pen. Thin Realism is a metaphysical thesis. It has nothing to do at all with whether the justification of an axiom references set theory alone or mathematics more generally. In fact, Pen’s Thin Realist does reference other areas of mathematics!
- You go on to talk of three notions of truth in set theory and you
say that we should just proceed with all three. This is something that has been discussed at length in the literature of pluralism in mathematics. The point I want to make here is that it requires an argument. You cannot just say: “Let’s proceed with all three!” For
comparison imagine a similar claim with regard to number theory or physics. One can’t just help oneself to relativism. It requires an argument!
For some time now I have wanted to write more concerning your
program. But I still don’t have a grip on the X where X is your view
and at this stage I can only make claims of the form “If your view is
X then Y follows.” Moreover, as the discussion has proceeded my grip on X has actually weakened. And this applies not just to the
philosophical parts of X but also to the mathematical parts of X.
Let’s start with something where we can expect an absolutely clear and unambiguous answer: A mathematical question, namely, the question Hugh asked. Let me repeat it:
What is ? You wrote in your message of Sept 29:
The IMH# is compatible with all large cardinals. So is the
It would also be useful to have an answer to the second question I
asked. The version of you specified in your next message to me
on sept 29:
The (crude, uncut) is the statement that V is #-generated and
if a sentence with absolute parameters holds in a cardinal-preserving,
#-generated outer model then it holds in an inner model. It implies a
strong failure of CH but is not known to be consistent.
does not even obviously imply . Perhaps you meant, the above
together with ? Or something else?