# Re: Paper and slides on indefiniteness of CH

Dear Peter and Hugh,

Peter:

1. As I said, #-generation was not invented as a “fix” for anything. It was invented as the optimal form of maximality in height. It is the limit of the small large cardinal hierarchy (inaccessibles, Mahlos, weak compacts, $latex\omega$-Erdos, $(\omega+\omega)$-Erdos, … #-generation). A nice feature is that it unifies well with the IMH, as follows: The IMH violates inaccessibles. IMH(inaccessibles) violates Mahlos. IMH(Mahlos) violates weak compacts … IMH(omega-Erdos) violates omega+omega-Erdos, … The limit of this chain of principles is the canonical maximality criterion $\textsf{IMH}^\#$, which is compatible with all small large cardinals, and as an extra bonus, with all large cardinals. It is a rather weak criterion, but becomes significantly stronger even with the tiny change of adding $\omega_1$ as a parameter (and considering only $\omega_1$ preserving outer models).

2. What I called “Magidor’s embedding reflection” in fact appears in a paper by Victoria Marshall (JSL 54, No.2). As it violates V = L it is not a form of height maximality (the problem is with the internal embeddings involved; if the embeddings are external then one gets a weak form of #-generation). Indeed Marshall Reflection does not appear to be a form of maximality in height or width at all.

You say: “I don’t think that any arguments based on the vague notion of ‘maximality’ provide us with much in the way of justification”. Wow! So you don’t think that inaccessibles are justified on the basis of reflection! Sounds like you’ve been talking to the radical Pen Maddy, who doesn’t believe in any form of intrinsic justification.

3. Here’s the most remarkable part of your message. You say:

“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.”

In this thread I have repeatedly and without objection taken Pen’s Thin Realism to be grounded on “good set theory” (or if looking beyond set theory, on “good mathematics”). So you have now rejected not only the HP, but also Thin Realism. My view is that Pen got it exactly right when it comes to evidence from the practice of set theory, one must only acknowledge that such evidence is limited by the lack of consensus on what “good set theory” means.

You are right to say that there is value to “predictions” and “verifications”. But these only serve to make a “good set theory” better. They don’t really change much, as even if a brand of “good set theory” fails to fulfill one of its “predictions”, it can still maintain its high status. Take the example of Forcing Axioms: They have been and always will be regarded as “good set theory”, even if the “prediction” that you attribute to them fails to materialise.

Peter, your unhesitating rejection of approaches to set-theoretic truth is not helpful. You faulted the HP for not being “well-grounded” as its grounding leans on a consensus regarding the “maximality of V in height and width”. Now you fault Thin Realism (TR) for not being “well-grounded” as its grounding leans on “good set theory”. There is an analogy between TR and the HP: Like Pen’s second philosopher, Max (the Maximality Man) is fascinated by the idea of maximality of V in height and width and he “sets out to discover what the world of maximality is like, the range of what there is to the notion and its various properties and behaviours”. In light of this analogy, it is reasonable that someone who likes Thin Realism would also like the HP and vice-versa. It seems that you reject both, yet fail to provide a good substitute. How can we possibly make progress in our understanding of set-theoretic truth with such skepticism? What I hear from Pen and Hugh is a “wait and see” attitude, they want to know what criteria and consensus comes out of the HP. Yet you want to reject the approach out of hand. I don’t get it. Are you a pessimist at heart?

It seems to me that in light of your rejection of both TR and HP, the natural way for you to go is “radical skepticism”, which denies this whole discussion of set-theoretic truth in the first place. (Pen claimed to be a radical skeptic, but I don’t entirely believe it, as she does offer us Thin Realism.) Maybe Pen’s Arealism is your real cup of tea?

OK, let’s return to something we agree on: the lack of consensus regarding “good set theory”, where I have something positive to offer. What this lack of consensus suggests to me is that we should seek further clarification by looking to other forms of evidence, namely Type 2 evidence (what provides the best foundation for math) and Type 3 evidence (what follows from the maximality of V in height and width). The optimistic position (I am an optimist at heart) is that the lack of consensus based solely on Type 1 evidence (coming from set-theoretic practice) could be resolved by favouring those Type 1 axioms which in addition are supported by Type 2 evidence, Type 3 evidence, or both. Forcing Axioms seem to be the best current axioms with both Type 1 and Type 2 support, and perhaps if they are unified in some way with Type 3 evidence (consequences of Maximality) one will arrive at axioms which can be regarded as true. This may even give us a glimmer of hope for resolving CH. But of course that is way premature, as we have so much work to do (on all three types of evidence) that it is impossible to make a reasonable prediction at this point.

To summarise this part: Please don’t reject things out of hand. My suggestion (after having been set straight on a number of key points by Pen) is to try to unify the best of three different approaches (practice, foundations, maximality) and see if we can make real progress that way.

4. With regard to your very entertaining story about K and Max: As I have said, one does not need a radical potentialist view to implement the HP, and I now regret having confessed to it (as opposed to a single-universe view augmented by height potentialism), as it is easy to make a mistake using it, as you have done. I explain: Suppose that “we live in a Hyperuniverse” and our aim is to weed out the “optimal universes”. You suggest that maximality criteria for a given ctm M quantify over the entire Hyperuniverse (“Our quantifiers range over CTM-space.”). This is not true and this is a key point: They are expressible in a first-order way over Goedel lengthenings of M. (By Gödel lengthening I mean an initial segment of the universe L(M) built over M, the constructible universe relative to M.) This even applies to #-generation, as explained below to Hugh. From the height potentialist / width actualist view this is quite clear (V is not countable!) and the only reason that Maximality Criteria can be reflected into the Hyperuniverse (denote this by H to save writing) is that they are expressible in this special way (a tiny fragment of second order set theory). But the converse is false, i.e., properties of a member M of H which are expressible in H (essentially arbitrary second-order properties) need not be of this special form. For example, no height maximal universe M is countable in its Goedel lengthenings, even for a radical potentialist, even though it is surely countable in the Hyperuniverse. Briefly put: From the height potentialist / width actualist view, the reduction to the Hyperuniverse results in a study of only very special properties of ctm’s, only those which result from maximality criteria expressed using lengthenings and “thickenings” of V via Löwenheim-Skolem.

So I was too honest, I should not have admitted to a radical form of multiversism (radical potentialism), as it is then easy to misundertand the HP as you have. As far as the choice of maximality criteria, I can only repeat myself: Please be open-minded and do not prejudge the programme before seeing the criteria that it generates. You will see that our intuitions about maximality criteria are more robust than our intuitions about “good set theory”.

Hugh:

1. The only method I know to obtain the consistency of the maximality criterion I stated involves Prikry-like forcings, which add Weak Squares. Weak Squares contradict supercompactness. In your last mail you verify that stronger maximality criteria do indeed violate supercompactness.

2. A synthesis of LCs with maximality criteria makes no sense until LCs themeselves are derived from some form of maximality of V in height and width.

3. I was postponing the discussion of the reduction of #-generation to ctm’s (countable transitive models) as long as possible as it is quite technical, but as you raised it again I’ll deal with it now. Recall that in the HP “thickenings” are dealt with via theories. So #-generation really means that for each Gödel lengthening $L_\alpha(V)$ of $V$, the theory in $L_\alpha(V)$ which expresses that $V$ is generated by a presharp which is $\alpha$-iterable is consistent. Another way to say this is that for each $\alpha$, there is an $\alpha$-iterable presharp which generates $V$ in a forcing extension of $L(V)$ in which $\alpha$ is made countable. For ctm’s this translates to: A ctm $M$ is (weakly) #-generated if for each countable $\alpha$, $M$ is generated by an $\alpha$-iterable presharp. This is weaker than the cleaner, original form of #-generation. With this change, one can run the LS argument and regard $\textsf{IMH}^\#$ as a statement about ctm’s. In conclusion: You are right, we can’t apply LS to the raw version of $\textsf{IMH}^\#$, essentially because #-generation for a (real coding a) countable $V$ is a $\Sigma^1_3$ property; but weak #-generation is $\Pi^1_2$ and this is the only change required.

But again, there is no need in the HP to make the move to ctm’s at all, one can always work with theories definable in Gödel lengthenings of V, making no mention of countability. Indeed it seems that the move to ctm’s has led to unfortunate misunderstandings, as I say to Peter above. That is why it is quite inappropriate, as you have done on numerous occasions, to refer to the HP as the study of ctm’s, as there is no need to consider ctm’s at all, and even if one does (by applying LS), the properties of ctm’s that results are very special indeed, far more special than what a full-blown theory of ctm’s would entail.