# Set-theoretical multiverses and their representation as functors? Why *the* multiverse?

In some related MO questions like The set-theoretic multiverse as a (bi)category it is discussed how one might represent the multiverse (see The set-theoretic multiverse) in a category theoretic way, e.g. by a category having universes as objects (models of ZFC). However it seems not so obvious which morphisms to take for such an approach (e.g. forcing relation or large cardinal embeddings or any other techniques producing new models from old ones).

Question 1. What implications do the different choices for a method to produce new set theoretic worlds from old ones have w.r.t. the multiverse / is there a right choice (what are the criteria)? If not, then the unfinished discussion about the right morphisms in a category theoretic representaation of a multiverse seems to just reflect this?

Question 2. However, is it essential in this context to talk about the multiverse rather than a multiverse? If it is not obvious which morphisms could be best, could't one regard different multiverses or multiverse representations and treat them equally until one finds justification to single out some i.e. the real multiverse?

Question 3. Couldn't it be the better way to think of a multiverse as a functor $$V: T\rightarrow M$$ mapping extensions of ZFC in $T$ to ZFC-models in $M$, after all this seems to be natural, since a multiverse is about variable worlds (and then think about multiverses with certain properties like having a left adjoint $V_*:M\rightarrow T$ or to study the functor category of multiverses $M^T$ or the like)?

• In your question 3, what is T? For that matter, what is M? – David Roberts Jul 30 '18 at 22:16

Of course we have been investigating a wide variety of multiverse concepts, and in this sense, yes, we have not just one, but many, multiverses.

But to be sure, much of this multiverse analysis has been inspired by the philosophical idea that there is or might be a unique grand multiverse for mathematics: the actual multiverse, but aware of the possibility that this is a mirage. Even when one has the idea, as many of us do, of looking for mathematical truth in various set-theoretic universes or categories, there might remain the question: which such universe or categories are there? What kind of universes can there be? To assert that there are facts-of-the-matter about what kind of universes exist is to take a step in the direction of the multiverse, and this is philosophical issue at stake in the singularity or plurality of the multiverse. Of course it is not satisfactorily answered simply by adopting the multiverse position itself.

I was led in my multiverse paper, to which you have linked, by this kind of thinking to formulate axioms expressing what kind of existence principles we would want or expect in such a multiverse.

How nice it was to observe that the philosophical issues transform into purely mathematical questions, when one proposes various specific mathematical multiverse conceptions and begins to analyze their mathematical nature. Thus, one gains philosophical insight by means of a purely mathematical investigation. And this investigation has been undertaken in earnest.

So the basic approach has been to study various specific multiverse conceptions as toy multiverses of a kind, standing in for the actual larger multiverse that we seek to understand.

Let me mention several instances.

Victoria Gitman and I looked at the multiverse consisting of the countable computably saturated models of set theory, in our paper:

This collection of models, it turns out, fulfill all of the multiverse axioms I had identified in my multiverse paper, and more.

Woodin had defined the generic multiverse of a model of set theory, which is the smallest collection containing that model and closed under forcing extensions and ground models. This is, of course, a robust multiverse conception, but nevertheless, there are many set-theoretic principles, such as the GCH, that are achievable over a model of set theory, but not necessarily by set forcing.

Benedikt Löwe and I had investigated the generic multiverse as a Kripke model with two natural modal operators, an upward-oriented forcing possibility and a downward-oriented ground-model possibility, and we explored the modal validities of this system in our papers on the modal logic of forcing and also in:

The Vienna approach, advanced by Sy Friedman and his collaborators, including Carolin Antos and others, has focused on the collection of countable transitive models of ZFC, the set-theoretic hyperverse, which is a multiverse conception, in which the concept of well-foundedness is emphasized as absolute. Friedman has identified a program of using the hyperverse as a means of identifying natural set-theoretic axioms, such as the Inner Model Hypothesis.

The New York approach, in contrast, in work of myself, Victoria Gitman and Kameryn Williams, has emphasized the non-absolute character of well-foundedness, and has accordingly accommodated ill-founded models of set theory. There have been many recent developments in this area, including the universal finite set:

Recently, in my work with Øystein Linnebo and others, we have explored diverse concepts of set-theoretic potentialism, which I view as multiverse perspectives.

In that paper, we look at various multiverse conceptions of potentialism, exploring collections of models of set theory under the relation of rank-extension and end-extension and many others, investigating in each case the modal validities of that potentialist conception.

In summary, there is a huge effort currently underway to investigate diverse multiverse conceptions.

Lastly, I notice that you mention the category-theoretic approach to the multiverse, so let me mention that, unfortunately, the category-theoretic perspective has not seemed to play a large role in recent work in this area. Perhaps ideas arising from this perspective will find a fruitful application in the future. I am hopeful about that.

• Is it still the New York approach? Or should we now call it the Oxford approach? – Kameryn Williams Jul 31 '18 at 2:42
• Oh dear, Kameryn. It seems one can never truly leave New York, and I shall leave a large-cardinal part of my heart here. So let us continue to call it the New York approach. – Joel David Hamkins Jul 31 '18 at 3:05
• Oh yes, I cite their paper in my paper on arithmetic potentialism and the universal algorithm jdh.hamkins.org/…. – Joel David Hamkins Jul 31 '18 at 3:42
• Thank you very much for explaining some of the background and different approaches to these ideas! You once wrote (if I understand well) that you favour settings where mathematical developments are driven by philosophical questions and the mathematical insights in turn lead to new philosophical insights and questions - could you recommend a reference where you explain more about this interaction between philosophical and mathematical developments - in this context or more generally? – FWE Jul 31 '18 at 10:41
• @FWE That kind of interaction seems to have been a recurring feature with some of my work, such as my recent work on potentialism (for example, see the diagram at the top of my notes for this talk: jdh.hamkins.org/…). The main potentialist idea originates classically and is philosophical; Linnebo made a turn toward mathematics by introducing the modal account, and this has now been much more developed mathematically, for we determined the precise modal commitments of various potentialist conceptions. – Joel David Hamkins Jul 31 '18 at 11:27