# Re: Paper and slides on indefiniteness of CH

Dear Sy,

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
more carefully.

1. 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.
2. 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!

1. 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 $\textsf{SIMH}^\#(\omega_1)$? You wrote in your message of Sept 29:

The IMH# is compatible with all large cardinals. So is the $\textsf{SIMH}^\#(\omega_1)$

It would also be useful to have an answer to the second question I
asked. The version of $\textsf{SIMH}^\#$ you specified in your next message to me
on sept 29:

The (crude, uncut) $\textsf{SIMH}^\#$ 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 $\textsf{IMH}^\#$. Perhaps you meant, the above
together with $\textsf{IMH}^\#$? Or something else?

Best,
Peter

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

you are ignoring the obvious; I explain.

this discussion is not about the philosophy of mathematics! It is about the concept of set, i.e. part of the philosophy of set theory! The obvious point you ignore is that there is a philosophy of set theory, independent of what is done in mathematics

I am claiming that the HP is a legitimate contribution to the philosophy of the set-concept, part of the philosophy of set theory. Of course there is more to the philosophy of set theory than just the philosophy of the set-concept, namely the philosophy of what is done in set theory as a branch of mathematics, but that is a different thing.

Well, as it happens, I do take set theory to be part of mathematics — and the philosophy of set theory to be part of the philosophy of mathematics.  You insist that you’re doing a kind of philosophy of set theory that isn’t part of philosophy of mathematics, and I reply that you must be attempting a short of pure philosophy, like the study of radical skepticism or a priori metaphysics.

OK!  So we have an answer to the question Peter has been asking:  are you an actualist or a potentialist?  Answer:  a potentialist.

Finally. I thought that this was already implicit (although not explicit) when you quoted Tatiana and I in your first e-mail to this group:

“Sy says some interesting things in his BSL paper about ‘true in V':  it doesn’t ‘reflect an ontological state of affairs concerning the universe of all sets as a reality to which existence can be ascribed independently of set-theoretic practice’, but rather ‘a façon de parler that only conveys information about set-theorists’ epistemic attitudes, as a description of the status that certain statements have or are expected to have in set-theorist’s eyes’ (p. 80).”

It is hard for me to imagine the above in conjunction with an actualist position.

I misunderstood this passage from your BSL paper.  There’s often a kind of translation problem when philosophical terms get used and we professional philosophers assume they’re intended in more or less the same sense as we use them.  In this case, the troublesome term is ‘ontological’.

I took you to be denying that the subject matter of set theory is an objectively existing abstract realm, not to be saying anything at all about how we should best think of V (as completed or potential) while doing set theory.

So now we have to understand better what you are out to do, which means understanding better what these ‘thickenings’ are supposed to be.

I realize that you and Hugh have been discussing this at a very high level, but at some point could you give one explicit HP-generated mathematical principle that you endorse, and explain its significance to us philosophers?

All best,
Pen

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

But haven’t you softened your position as well? You went from:

“I don’t see why anyone has reason to sign onto this project, or to care about it one way or the other, unless it reveals some mathematical interest despite you.”

to:

“my strongest protest — why should mathematicians care about a program that doesn’t even aim to produce any good mathematics? — is no longer valid.  And my interest revives …”

You changed the word “anyone” to “mathematicians”! Of course I agree that mathematicians shouldn’t care unless there is “good math” coming out! My disappointment was with your earlier version, which implied that not even *philosophers* should care unless there is “good math” coming out! That I took as a denunciation of the HP on purely philosophical grounds.

As it happens, I had typed ‘anyone’ in my recent message, then changed it to ‘mathematicians’ in hope of sharpening the point!  Anyway, your suggestion is that some one might be interested in the program as philosophy, not as mathematics.  Well, I suppose that’s possible.  But it looks to me as if the kind of philosophy in question would have to be a kind of pure philosophy, like the study of radical skepticism or analytic metaphysics.  It wouldn’t be philosophy of mathematics, because philosophy of mathematics is about what’s doneas mathematics.  To make an extreme comparison (swiped from Wittgenstein), some one might be interested in the program because it generates such attractive arrays of symbols, so attractive that they make nice wallpaper patterns.  I was assuming your program is a mathematical one, not an exercise in pure philosophy or interior decoration.

So here I was intrigued by your exchange with Peter.  But I blush to admit that I still haven’t grasped your answer to the flat-footed question:  if there is no actual V, in width or height, what are we asking about when we ask about CH?

Finally you ask an easy question! (Your other questions were all very challenging.)

Answer to this question: We have many pictures of V. Through a process of comparison we isolate those pictures which best exhibit the feature of Maximality, the “optimal” pictures. Then we have 3 possibilities:

a. Does CH hold in all of the optimal pictures?
b. Does CH fail in all of the optimal pictures?
c. Otherwise

In Case a, we have inferred CH from Maximality, in Case b we have inferrred -CH from Maximality and in Case c we come to no definitive conclusion about CH on the basis of Maximality.

OK!  So we have an answer to the question Peter has been asking:  are you an actualist or a potentialist?  Answer:  a potentialist.  So you aren’t really out to settle CH in the ordinary way people think of that project; you aren’t out to discover new things about V (because there is no V).

So now we have to understand better what you are out to do, which means understanding better what these ‘thickenings’ are supposed to be.  Is this what Harvey has been trying to pin down?

(Let me again suggest that it might be worth reconstituting the list so that people can remove themselves in private.)

I still think that this discussion will soon “fizzle out” but I thought that before and was proved wrong. So I think you have a good idea. Do you know how to do that? I am not clever with “e-mail news groups”; can someone volunteer to set things up so that each of us can subscribe or unsubscribe freely? As it currently stands, someone will write to me occasionally and ask to be left out, but then I have to police the situation to make sure that future mails don’t go to that person; not an ideal setup.

This discussion will eventually fizzle out, but it occurs to me that there may be (and indeed have been) other questions of interest to this group that could benefit from discussion in this forum.  But maybe not.

In any case, no, I don’t know how to set these things up.

All best,
Pen