# Re: Paper and slides on indefiniteness of CH

Dear Sy,

I think I now have a much better grip on the picture you are working with. This letter is an attempt to sum thing up — both mathematical and philosophical — and express my misgivings, in what I hope you will take to be a light-hearted and friendly manner.

In my letter on Oct. 26 I carefully laid out the varieties of potentialism and actualism and asked which version you held. You answered on Oct. 26 and I was pretty sure that I understood. But I wanted to be sure so I asked for confirmation. You confirmed that in your P.S. (of a letter on a different topic) on Oct. 26:

PS: In answer to an earlier question, I am indeed naturally inclined to think in terms of the stronger form of radical potentialism. Indeed I do think that, as with height actualism, there are arguments to suggest that the weaker form of radical potentialism without the stronger form is untenable.

Here “the stronger form of radical potentialism” was the one I explicitly asked for confirmation on. To be clear: You endorse the strong form of radical potentialism according to which for every transitive model of ZFC there is an actual extension (meaning an actual lengthening and/or thickening) in which that model is actually seen to be countable.

So, on this view, everything is ultimately (and actually) countable. Thus, on this view, we actually live in the hyperuniverse, the space of countable transitive models of ZFC. That’s all there is. That’s our world.

This is very close to Skolem’s view. He took it to entail that set theory had evaporated, which is why I used that phrase. But you do not. Why? Because one can still do set theory in this limited world. How?

This brings us back to my original questions about your “dual use of ‘V'”, at times for little-V’s (countable transitive models of ZFC) and at other times for “the real thing”, what I called SUPER-V (to disambiguate the notation). I had originally thought that your view was this: There is SUPER-V. Actualism holds with regard to SUPER-V. There are no actual lengthenings or thickenings of SUPER-V, only virtual ones. Everything takes place in SUPER-V. It is the guide to all our claims about “intrinsic justifications on the basis of the “maximal” iterative conception of set” (subsequently demoted to “intrinsically motivated (heuristically) on the basis of the ‘maximal’ iterative conception of set”). By appealing to the downward Löwenheim-Skolem theorem I thought you were arguing that without loss of generality we could gain insight into SUPER-V by investigating the space of countable transitive models of ZFC.

The virtue of that picture (which I erroneously thought you held) is that you would have something to hang your hat on — SUPER-V –, something to cash out the intuitions that you claimed about the ” ‘maximal’ iterative conception of set”. The drawback what that it was hard to see (for me at least) how we could gain insight into SUPER-V (which had no actual lengthenings or thickenings) by investigating countable transitive models of ZFC (which do!).

But that is all neither here not there, since that is not your view. Your view is far more radical. There is just the hyperuniverse, the space of countable transitive models of ZFC. There is no need for appeal to the Löwenheim-Skolem theorem, since everything is countable!

I now have a much better grip on the background philosophical picture. This is what I suspected all along the way that is why I have been pressing you on these matters.

I want now to examine this world view — to take ii seriously and elaborate its consequences. To do that I will follow your lead with Max and tell a story. The story is below, out of the main body of this letter.

Best,
Peter

Let me introduce K. He has a history of getting into situations like this.

Let us enter the hyperuniverse…

K awakes. He looks around. He is surrounded by countable transitive models of ZFC. Nothing else.

How did I get here? Why are all these countable transitive models of ZFC kicking around? Why not just countable transitive models of ZFC – Replacement + $\Sigma_2$-Replacement? Why not anything else?

K takes a stroll in this strange universe, trying to get his bearing. All of the models he encounters are transitive models of ZFC. He encounters some that satisfy $V = L$, some that satisfy $\textsf{PD}$, some that satisfy $\textsf{MM}^{++}$, etc. But for every model he encounters he finds another model in which the previous model is witnessed to be countable.

He thinks: “I must be dreaming. I have fallen through layers of sleep into the world of countable transitive models of ZFC. The reason all of these countable transitive models of $V = L$, $\textsf{PD}$, $\textsf{PFA}$, etc. are kicking around is that these statements are $\beta$-consistent, something I know from my experience with the outer world. In the outer world, before my fall, I was not wedded to the idea that there was a SUPER-$V$ — I was open minded about that. But I was confident that there was a genuine distinction between the countable and the uncountable. And now, through the fall, I have landed in the world of the countable transitive models of ZFC. The uncountable models are still out there — everything down here derives from what lies up there.”

At this point a voice is heard from the void…

S: No! You are not dreaming — you have not fallen. There is no outer world. This is all that there is. Everything is indeed countable.

K: What? Are you telling me set theory has evaporated.

S: No. Set theory is alive and well.

K: But all of the models around here are countable, as witnessed by other models around here. That violates Cantor’s theorem. So, set theory has evaporated.

S: No! Set theory is alive and well. You must look at set theory in the right way. You must redirect your vision. Attend not to the array of all that you see around you. Attend to what holds inside the various models. After all, they all satisfy ZFC — so Cantor’s Theorem holds. And you must further restrict your attention, not just to any old model but to the optimal ones.

K: Hold on a minute. I see that Cantor’s Theorem holds in each of the models around here. But it doesn’t really hold. After all, everything is countable!

S: No, no, you are confused.

[K closes his eyes...]

S: Hey! What are you doing?

K: I’m trying to wake up.

S: Wait! Just stay a while. Give it a chance. It’s a nice place. You’ll learn to love it. Let me make things easier. Let me introduce Max. He will guide you around.

[Max materializes.]

Max takes K on a tour, to all the great sites — the “optimal” countable transitive models of ZFC. He tries to give K a sense of how to locate these, so that one day he too might become a tour guide. Max tells K that the guide to locating these is “maximality”, with regard to both “thickenings” and “lengthenings”.

K: I see, so like forcing axioms (for “thickenings”) and the resemblance principles of Magidor and Bagaria (for “lengthenings”)?

Max: No, no, not that. The “optimal” models are “maximal” in a different sense. Let me try to convey this sense.

Let’s start with IMH. But bear in mind this is just a first approximation. It will turn out to have problems. The goal is to investigate the various principles that are suggested by “maximality” (as a kind of “intrinsic heuristic”) in the hope that we will achieve convergence and find the true principles of “maximality” that enable us to locate the “optimal” universes. O

[Insert description of IMH. Let "CTM-Space" be the space of transitive models of ZFC. In short, let "CTM-Space" be the world that K has fallen into.]

K: I see why you said that there would be problems with IMH: If $V\in \text{CTM-Space}$ satisfies IMH, then $V$ contains a real $x$ such that $V$ satisfies “For every transitive model $M$ of ZFC, $x$ is not in $M$“; in particular, there is no rank initial segment of $V$ that satisfies ZFC and so such a $V$ cannot contain an inaccessible cardinal. In fact, every ordinal of such a $V$ is definable from $x$. So such a $V$ is “humiliated” in a dramatic fashion by a real $x$ within it.

Max: I know. Like I said, it was just a first approximation. IMH is, as you observe, incompatible with inaccessible cardinals in a dramatic way. I was just trying to illustrate the sense of “width maximality” that we are trying to articulate. Now we have to simultaneously incorporate “height maximality”. We do this in terms of #-generation.

[Insert description of #-generation and $\textsf{IMH}^\#$]

K: I have a bunch of problems with this. First, I don’t see how you arrive at #-generation. You use the # to generate the model but then you ignore the #.

Second, there is a trivial consistency proof of $\textsf{IMH}^\#$ but it shows even more, namely, this: Assume that for every real $x,$ $x^\#$ exists. Then there is a real $x_0$ such that for any $V\in \text{CTM-Space}$ containing $x_0$, $V$ satisfies Extreme-$\textsf{IMH}^\#$ in the following sense: if $\varphi$ holds in any #-generated model $N$ (whether it is an outer extension of V or not) then $\varphi$ holds in an inner model of $V$. So what is really going on has nothing to do with outer models — it is much more general than that. This gives not just the compatibility of $\textsf{IMH}^\#$ with all standard large cardinal but also with all choiceless large cardinals.

[It should be clear at this point (and more so below) that despite this new and strange circumstances K is still able to access H.]

Third, there is a problem of articulation. The property of being a model $V\in \text{CTM-Space}$ which satisfies $\textsf{IMH}^\#$ is a $\Pi_3$ property over the entire space $\text{CTM-Space}$. When I was living in the outer world (where I could see $\text{CTM-Space}$ as a set) I could articulate that property and thereby locate the V’s in CTM-Space that satisfy $\textsf{IMH}^\#$. But how can you (we) do that down here? If you really believe that the $\Pi_3$ property over the space $\text{CTM-Space}$ is a legitimate property then you are granting that the domain $\text{CTM-Space}$ is a determinate domain (to make sense of the determinateness of the alternating quantifiers in the $\Pi_3$ property). But if you believe that $\text{CTM-Space}$ is a determinate domain then why can’t you just take the union of all the models in $\text{CTM-Space}$ to form a set? Of course, that union will not satisfy ZFC. But my point is that by your lights it should make perfect sense, in which case you transcend this world. In short you can only locate the models $V\in \text{CTM-Space}$ that satisfy $\textsf{IMH}^\#$ by popping outside of $latex \text{CTM-Space}$!

Max: Man, are you ever cranky…

K: I’m just a little lost and feeling homesick. What about you? How did you get there? I have the sense that you got here they same way I did and that (in your talk of $\textsf{IMH}^\#$) you still have one foot in the outer world.

Max: No, I was born here. Bear with me. Like I said, we are just getting started. Let’s move on to $\textsf{IMH}^\#$.

K: Wait a second. Before we do that can you tell me something?

Max: Sure.

K: We are standing in  $\text{CTM-Space}$, right?

Max: Right.

K: And from this standpoint we are proving things about various principles that hold in various $V$‘s in $\text{CTM-Space}$, right?

Max: Right.

K: What theory are allowed to use in proving these results about things in $\text{CTM-Space}$? We are standing here, in $\text{CTM-Space}$. Our quantifiers range over $\text{CTM-Space}$. Surely we should be using a theory of $\text{CTM-Space}$. But we have been using ZFC and that doesn’t hold in $\text{CTM-Space}$. Of course, it holds in every V in $\text{CTM-Space}$ but it does not hold in $\text{CTM-Space}$ itself. We should be using a theory that holds in $\text{CTM-Space}$ if we are to prove things about the objects in $\text{CTM-Space}$. What is that theory?

Max: Hmm … I see the point … Actually! … Maybe we are really in one of the $V$‘s in $\text{CTM-Space}$! This $\text{CTM-Space}$ is itself in one of the $V$‘s of a larger $\text{CTM-Space}$

K (to himself): This is getting trippy.

Max: … Yes, that way we can invoke ZFC.

K: But it doesn’t make sense. Everything around us is countable and that isn’t true of any $V$ in any $\text{CTM-Space}$.

Max: Good point. Well maybe we are in the $\text{CTM-Space}$ of a $V$ that is itself in the $\text{CTM-Space}$ of a larger $V$.

K: But that doesn’t work either. For then we can’t help ourself to ZFC. Sure, it holds in the $V$ of whose $\text{CTM-Space}$ we are locked in. But it doesn’t hold here! You seem to want to have it both ways — you want to help yourself to ZFC while living in a $\text{CTM-Space}$.

Max: Let me get back to you on that one … Can we move on to $\textsf{SIMH}^\#$.

K: Sure.

[Insert a description of the two version of $\textsf{SIMH}^\#$. Let us call these Strong-$\textsf{SIMH}^\#$and Weak-$\textsf{SIMH}^\#$. Strong-$\textsf{SIMH}^\#$ is based on the unification of $\textsf{SIMH}$ (as formulated in the 2006 BSL paper) but where one restricts to #-generated models. (See Hugh's letter of 10/13/14 for details.) Weak-$\textsf{SIMH}^\#$ is the version where one restricts to cardinal-preserving outer models.]

K: Well, Strong-$\textsf{SIMH}^\#$ is not know to be consistent. It does imply $\textsf{IMH}^\#$ (so the `S’ makes sense). But it also strongly denies large cardinals. In fact, it implies that there is a real $x$ such that $\omega_1^{L[x]}=\omega_1$ and hence that $x^\#$ doesn’t exist! So that’s no good.

Weak-$\textsf{SIMH}^\#$ is not known to be consistent, either. Moreover, it is not known to imply $\textsf{IMH}^\#$ (so why the “S”). It is true that it implies not-CH (trivially). But we cannot do anything with it since since very little is known about building cardinal-perserving outer models over an arbitrary initial model.

Max: Like I said, we are just getting started.

[Max goes on to describe various principles concerning HOD that are supposed to follow from "maximality". K proves to be equally "difficult".]

K: Ok, let’s back up. What is our guide? I’ve lost my compass. I don’t have a grip on this sense of “maximality” that you are trying to convey to me. If you want to teach me how to be a tour guide and locate the “optimal” models I need something to guide me.

Max: You do have something to guide you, namely, the “maximal” iterative conception of set”!

K: Well, I certainly understand the “iterative conception of set” but when we fell into $\text{CTM-Space}$ we gave up on that. After all, every model here is witnessed to be countable in another model. Everything is countable! That flies in the face of the “iterative conception of set”, a conception that was supposed to give us ZFC, which doesn’t hold here in $\text{CTM-Space}$.

Max: No, no. You are looking at things the wrong way. You are fixated on $\text{CTM-Space}$. You have to direct your attention to the models within it. You have to think about things differently. You see, in this new way of looking at things to say that a statement $\varphi$ is true (in this new sense) is not to say that it holds in some $V$ in $\text{CTM-Space}$; and it is not to say that it holds in all $V$‘s in $\text{CTM-Space}$; rather it is to say that it holds in all of the “optimal” $V$‘s in $\text{CTM-Space}$. This is our new conception of truth: We declare $\varphi$ to be true if and only if it holds in all of the “optimal” V’s in $\text{CTM-Space}$. For example, if we want to determine whether CH is true (in this new sense) we have to determine whether it holds in all of the “optimal” $V$‘s in $\text{CTM-Space}$. If it holds in all of them, it is true (in this new sense), if it fails in all of them it is false (in this new sense), and if it holds in some but not in others then it is neither true nor false (in this new sense). Got it?

K: Yeah, I got it. But you are introducing deviant notions. This is no longer about the “iterative conception of set” in the straightforward sense and it is no longer about truth in the straightforward sense. But let me go along with it, employing these deviant notions and explaining why I think that they are problematic.

It was disconcerting enough falling into $\text{CTM-Space}$. Now you are asking me to fall once again, into the “optimal” models in $\text{CTM-Space}$. You are asking me to, as it were, “thread my way” through the “optimal” models, look at what holds across them, and embrace those statements at true (in this new, deviant sense).

I have two problems with this:

First, this whole investigation of principles — like $\textsf{IMH}$, $\textsf{IMH}^\#$, Strong-$\textsf{SIMH}^\#$, Weak-$\textsf{IMH}^\#$, etc. — has taking place in $\text{CTM-Space}$. (We don’t have ZFC here but we are setting that aside. You are going to get back to me on that.) The trouble is that you are asking me to simultaneously view things from inside the “optimal” models (to “thread my way through the ‘optimal’ models”) via principles that make reference to what lies outside of those models (things like actual outer extensions). In order for me to make sense of those principles I have to occupy this external standpoint, standing squarely in $\text{CTM-Space}$. But if I do that then I can see that none of these “optimal” models are the genuine article. It is fine for you to introduce this deviant notion of truth — truth (in this sense) being what holds across the “optimal” models. But to make sense of it I have to stand right here, in $\text{CTM-Space}$, and access truth in the straightforward sense — truth in $\text{CTM-Space}$. And those truths (the ones required to make sense of your principles) undermine the “optimal” models since, e.g., those truths reveal that the “optimal” models are countable!

But let us set that aside. (I have set many things aside already. So why stop here.) There is a second problem. Even if I were to embrace this new conception of truth — as what holds across the “optimal” models in $\text{CTM-Space}$ — I am not sure what it is that I would be embracing. For this new conception of truth makes reference to the notion of an “optimal” model in $\text{CTM-Space}$ and that notion is totally vague. It follows that this new notion of truth is totally vague.

You have referred to a specific sense of “maximality” but I don’t have clear intuitions about the notion you have in mind. And the track record of the principles that you claimed to generate from this notion is, well, pretty bad, and doesn’t encourage me in thinking that there is indeed a clear underlying conception.

Tell me Max: How do you do it? How do you get around? What is your compass? How are you able to locate the “optimal” models? How are you able to get a grip on this specific notion of “maximality”?

Max: That’s easy. I just ask S!

[With those words, K awoke. No one knows what became of Max.]

THE END