Re: Paper and slides on indefiniteness of CH

Dear Pen and Hugh,

My apologies to my e-mail victims. This mail is intended for Pen and Hugh, but I am sending it to the whole group at the risk of offending some by its sheer length. It is much, much longer than I intended, but somehow the story went on and on and even this version is an abridged form of the more complete maximality story up to now. My aim is to try to communicate to Pen and Hugh, and perhaps others, how I see the HP, which is I think very different from the way they have regarded it.

Best wishes and appropriate apologies,
Sy

Please allow me to introduce my friend Max, also known as the Maximality Man. Max doesn’t have a PhD in Set Theory and has never proved a theorem, but he knows about first-order logic, knows what transitive models of ZFC are and how one can be a rank-initial segment or inner model of another. He is fascinated by the idea of Maximality for the universe of sets V 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” (sound familiar?). For this purpose he talks to set-theorists (the ones proving the theorems in set theory) and asks them naive (sometimes stupid) questions, hoping to find out what is possible, what is not possible and what is the best way to formulate the Maximality of V in mathematical terms. He has never heard of large cardinals, forcing or determinacy; but he knows about L and simple notions of definability like ordinal-definability.

Max starts with the question: I know that everything comes down to the ordinals and the powerset operation, which determine the “height” and “width” of V; can these be made Maximal?

The set-theorists reply: What do you mean by Maximal, Max?
In turn Max replies: Well, for example, I want V to be as fat as possible, is there a fattest possible V?

The set-theorists readily reply: Max, this is a stupid question because V is already as fat as possible, it has all of the sets; you can’t make it fatter! But now Max is confused. He replies: But someone told me that maybe V is the same as L; are you saying that you can’t even make L any fatter, even in your wildest imagination? Are you telling me that even L is as fat as possible?

The set-theorists didn’t expect Max to be so clever and reluctantly admit: OK, Max, here’s the deal. In fact we do have ways of enlarging countable little-V’s (countable transitive models of ZFC) by something called forcing and I suppose we could “imagine” a forcing extension of V itself, but don’t get too excited, because these forcing extensions don’t really exist, they are only in our imagination.

I see, says Max, so we can’t really make V fatter, we can only imagine it. And I suppose that making it fatter won’t make any difference anyway, as after fattening, V will look just about the same. Thanks for answering my question.

The set-theorists keep their mouths shut, hoping that Max will go away. But then they start to feel guilty and break the news to Max:

Well, Max, if V equals L then it would indeed make a big difference if we could fatten V, as L looks very special and after fattening it, its special appearance would be lost.

Now Max gets mad. Wait a minute, someone tells me that V could equal L and now you tell me that if this is the case then we are stuck in a universe that can only imagine all sorts of exciting stuff out there in its “imaginary fattenings” and not enjoy them face-to-face? And at the same time you tell me that V is already maximal, even if it equals L? You are deceiving me!

OK, Max, calm down. We didn’t say that V had to be L. It seems that you are not happy if it is, so let’s just agree that V is not L. Do you feel better now?

Max: A little better, but tell me, if V isn’t L then what is it? It seems to me that it should already be so “fat” so that further imaginary fattening wouldn’t produce new exciting stuff that V itself is missing. Can you give me a V like that? That’s where I want to put down roots and raise my family.

Now the set-theorists are confused. They are thinking: We’ve got to give Max a nice fat V which won’t be affected by fattening, but what is “fattening”? Let’s see if he will be happy with forcing.

Hey Max, we can give you what you want: An axiom which contradicts V = L and implies that if you fatten V with the best fattening method we know then V won’t change much. That method is called forcing.

Max: Thanks, that’s great! Now I know that I’ll never be missing anything in such a V! I am happy now … But Max is a worrier, so after a few days he comes back: Thanks for giving me a forcing-fat universe. But I just want to check: There aren’s other ways of fattening the universe, are there? I mean there aren’t other fancier ways of fattening that forcing does not provide, right?

The set-theorists are embarrassed. Well, Max, we didn’t want to mention that there is class forcing, hyperclass forcing, hyperhyperclass forcing, … and these are stronger ways to fatten V. But aren’t you happy just with plain old forcing?

But Max is greedy: I want my V to be unaffected by any conceivable fattening method! Isn’t there a strongest possible way to fatten V? Then if V is not affected by that strongest possible method then I’ll be happy.

The set-theorists can’t think of a strongest possible forcing method. Max is very disappointed. He decides to put this fattening business on hold and instead ask his 2nd question.

I want V to be as tall as possible, is there a tallest possible V? Again the set-theorists disappoint him: Max, this is another dumb question, of course V is already maximal in height, V has all of the ordinals! But Max was prepared for this, so he knows what to say: But someone told me that V can be so short that no V_alpha is a model of ZFC. Are you saying that even such a V is maximal in height? Can’t you at least imagine making such a V longer, to a taller model of ZFC?

This time the set-theorists don’t know what to say to Max. Some think that increasing the height of V makes no sense, whereas others can easily imagine it; indeed they agree with the following quote of Geoffrey Hellman:

The idea that any universe of sets can be properly extended (in height, not width) is extremely natural, endorsed by many mathematicians (e.g. MacLane, seemingly by Gödel, et. al.) … As Maddy and others say, if it’s possible that sets beyond some (putatively maximal) level exist, then they do exist …

Max finds out about this quote and announces that he will not take “no” for an answer: Universes can be lengthened! There is no “top” to the ordinals! He feels that as long as people don’t think he’s crazy he is going to take this view, which he holds deep in his heart.

Well, this story goes on. To fast forward: Max learns about reflection and more generally about maximality in height, and is ecstatic to learn that his worries about width maximality have disappeared: there is a simple way of expressing it, with no restriction on the meaning of fattening at all, using the lengthenings that he so loves. Things are looking pretty good, he has a nice mathematical notion of height and width maximality with respect to first-order sentences: If a first-order sentence holds in a height-maximal fattening of V then it holds in a thinning of V and if a first-order sentence with parameters holds in a “canonical lengthening” of V then it holds in a “canonical shortening” of V. [Essentially Sy's $\textsf{IMH}^\#$]

Max is still unsatisfied. The width maximality (with respect to fattenings) is only about sentences without parameters. Greedy Max wants width maximality with parameters too! He goes back to the set-theorists and pesters them again:

Why can’t we have width maximality with parameters? The set-theorists roll their eyes: You silly Max, the $\aleph_1$ of V can become countable in a fattening of V, this contradicts width maximality with parameters! Max replies: OK, then don’t let this happen, just consider fattenings which don’t make $\aleph_1$ countable! You silly Max, even so, the $\aleph_2$ of V can have size $\aleph_1$ in a fattening of V! Max gets the point. OK, then let’s only consider fattenings which preserve all of the cardinals! [Essentially Sy's version of the $\textsf{SIMH}^\#$]

But now Max starts to worry. Is this really width maximality, to only consider fattenings which preserve cardinalities? Maybe we should first choose our parameters, then only insist on preserving the cardinals we need to preserve to avoid a contradiction. [Essentially Sy's SIMH but with # added, i.e. what Hugh wanted]

But now something unexpected happens. They break the news to Max that the latter implies that $\aleph_1$ is small: For some real $x$, it is the same as the $\aleph_1$ of the skinny little model $L[x]$. Now Max is really confused. He thought he was making his way towards maximality and now finds that he is pretty close to the model $L$ which he was trying to escape from in the first place! He appeals to the set-theorists for help.

The set-theorists tell him that maybe his mistake is to start talking about preserving cardinals before maximising the notion of cardinal itself. In other words, maybe he should require that $\aleph_1$ is not equal to the $\aleph_1$ of $L[x]$ for any real $x$ and more generally that for no cardinal $\kappa$ is $\kappa^+$ equal to $(\kappa^+)^{L[A]}$ when $A$ is a subset of kappa. In fact maybe he should go even further and require this with $L[A]$ replaced by the much bigger model $\text{HOD}_A$ of sets hereditarily-ordinal definable with the parameter $A$! [Sy's Maximality Protocol, Part 2]

Max likes this idea but now the problem is that the set-theorists are facing challenges that they have never seen before. The forms of maximality that Max wants are so new that the set-theorists don’t yet have the tools to figure out which forms are consistent and which are not; also they want to keep Max happy. Max is just honestly trying to understand how maximality is best captured mathematically, but in the lengthy and complex process of discovery there are many wrong turns and disappointments, laced with some successes. Although the process exhibits genuine progress, converging closer and closer to the kind of maximality principle Max is looking for, the challenge of providing consistent forms of maximality that will answer Max’s fair questions is enormously difficult. We can’t blame Max! He is just a sincere and likable guy who has “set 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”. The set-theorists learn to like him and develop an equally sincere hope to fulfill his highest expectations.