Dear Peter,

Dear Sy,

You now appear to have endorsed width actualism. (I doubt that you actually believe it but rather have phrased your program in terms of width actualism since many people accept this.)

No, I have not endorsed width actualism. I only said that the HP can be treated equivalently either with width actualism or with radical potentialism.

Below is what I wrote to Geoffrey about this on 25.September:

Yes, in a nutshell what I am saying is that there are two “equivalent” ways of handling this:

1. You are a potentialist in height but remain actualist in width.

Then you need the quotes in “thickenings” when talking about width maximality. But if you happen to look at width maximality for a countable transitive model, then “thickening” = thickening without quotes (your really can thicken!).

2. You let loose and adopt a potentialist view for both length and width.

Then you don’t need to worry about quotes, because any picture of V is potentially countable, i.e. countable in some large picture V*, where all of the “thickenings” of V that you need are really thickenings of V that exist inside V*.

Mathematically there is no difference, but I do appreciate that philosophically one might feel uncomfortable with width potentialism and therefore wish to work with “thickenings” in quotes and not real thickenings taken from some bigger picture of V.

You said that for the purposes of the program one could either use length potentialism + width potentialism or length potentialism + width actualism. The idea, as I understand it, is that on the latter one has enough height to encode “imaginary width extensions”. I take it that you are then using the hyper universe (where one has actual height and width extensions) to model the actual height extensions and virtual width extensions of “V” (“the real thing”).

Is this right? Would you mind spelling out a bit for me why the two approaches are equivalent? In particular, I do not fully understand how you are treating “imaginary width extensions”.

Below is a copy of my explanation of this to Geoffrey on 24.September; please let me know if you have any questions. As Geoffrey pointed out, it is important to put “thickenings” in quotes and/or to distinguish non-standard from standard interpretations of power sets.

Dear Geoffrey,

Thanks for your valuable comments. It is nice to hear that you are happy with “lengthenings”; I’ll now try to convince you that there is no problem with “thickenings”, provided they are interpreted correctly. Indeed, you are right, “lengthenings” and “thickenings” are not fully analogous, there are important differences between these two notions, which I can address in a future mail (I don’t want this mail to be too long).

So as the starting point of this discussion, let’s take the view that V can be “lengthened” to a longer universe but cannot be thickened by adding new sets without adding new ordinals.

We can talk about forcing extensions of V, but we regard these as “non-standard”, not part of V. What other “non-standard extensions” of V are there? Surely they are not all just forcing extensions; what are they?

To answer this question it is very helpful to take a detour through a study of countable transitive models of ZFC. OK, I understand that we have dropped V for now, but please bear with me on this, it is instructive.

So let denote a countable transitive model of ZFC. What “non-standard extensions” does have? Of course just like , has its forcing extensions. A convenient fact is that forcing extensions of can actually be realised as new countable transitive models of ZFC; these are genuine thickenings of that exist in the ambient big universe big-V. This is not surprising, as is so little. But we don’t even have to restrict ourselves to forcing extensions of , we can talk about arbitrary thickenings of , namely countable transitive models of ZFC with the same ordinals as but more sets.

Alright, so far we have our together with all of its thickenings. Now we bring in Maximality. We ask the following question: How good a job does do of exhibiting the feature of Maximality? Of course we immediately say: Terrible! is only countable and therefore can be enlarged in billions of different ways! But we are not so demanding, we are less interested in the fact that can be enlarged than we are in the question of whether can be enlarged in such a way as to reveal new properties, new and interesting internal structures, …, things we cannot find if stay inside .

I haven’t forgotten that we are still just playing around with countable models, please bear with me a bit longer. OK, so let’s say that does a decent job of exhibiting Maximality if any first-order property that holds in some thickening of already holds in some thinning, i.e. inner model, of . That seems to be a perfectly reasonable demand to make of if is to be admitted to the Maximality Club. Please trust me when I say that there are such ‘s, exhibiting this form of Maximality. Good.

Now here is the next important observation, implicit in Barwise but more explicit in M.Stanley: Let be a countable transitive model of ZFC which lengthens little-V. There are little-V’s in the Maximality Club which have such lengthenings. (Probably this is not a big deal for you, as you believe that itself should have such a lengthening.) The interesting thing is this: Whether or not a given first-order property holds in a thickening of is something definable inside . More exactly, there is a logic called “-logic” which can be formulated inside and a theory T in that logic whose models are exactly the isomorphic copies of thickenings of ; moreover whether a first-order statement is consistent with T is definable inside . In summary, the question of whether a first-order property phi holds in a thickening of , a blatantly semantic question, is reduced to a syntactic question which can be answered definably inside : we just ask if $\varphi$ is consistent with the magic theory T. (Yes, this is a Completeness Theorem in the style of Gödel-Barwise.)

Very interesting. So if you allow to be lengthened, not even thickened, you are able to “see” what first-order properties hold in thickenings of and thereby determine whether or not belongs to the Maximality Club. This is great news, because now we can throw away our thickenings! We just talk about which first-order properties are consistent with our magic theory T and this is fuilly described in , any lengthening of to a model of ZFC. We don’t need real thickenings anymore, we can just talk about imaginary “thickenings”, i.e. models of T.

Thanks for your patience, now we go back to . You are OK with lengthenings, so let V^+ be a lengthening of V to another model of ZFC. Now just as for , there is a magic theory T described in whose models are the “thickenings” of , but now it’s “thickenings” in quotes, because these models are, like forcing extensions of , only “non-standard extensions” of V in the sense to which you referred. In we can define what it means for a first-order sentence to hold in a “thickening” of ; we just ask if it is consistent with the magic theory T. And finally, we can say that belongs to the Maximality Club if any first-order sentence which is consistent with T (i.e. holds in a “thickening” of ) also holds in a thinning (i.e. inner model) of . We have said all of this without thickening ! All we had to do was “lengthen” to a longer model of ZFC in order to understand what first-order properties can hold in “thickenings” of V.

So I hope that this clarifies how the IMH works. You don’t really need to thicken V, but only “thicken” , i.e. consider models of theories expressible in V^+. These are the “thicker pictures” that I have been talking about. And the IMH just says that V belongs to the Maximality Club in the above sense.

From a foundational and philosophical point of view the two pictures are quite different. On the first CH does not have fixed sense in a specific (upward-open-ended) V but instead one must look at CH across various “candidates for V”. On the second CH does have fixed sense in a specific (upward-open-ended) V. And, as far as I understand your implementation of the first approach for every candidate V there is an extension (in width and height) in which that candidate is countable, as in the case of the hyper universe. Is that right?

Yes. Below is what I said to Pen about this on 23.September:

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.

Best,

Sy