[bibtex key=Hamkins2016:UpwardClosureAndAmalgamationInTheGenericMultiverse]

**Abstract.** I prove several theorems concerning upward closure and amalgamation in the generic multiverse of a countable transitive model of set theory. Every such model $W$ has forcing extensions $W[c]$ and $W[d]$ by adding a Cohen real, which cannot be amalgamated in any further extension, but some nontrivial forcing notions have all their extensions amalgamable. An increasing chain $W[G_0]\subseteq W[G_1]\subseteq\cdots$ has an upper bound $W[H]$ if and only if the forcing had uniformly bounded essential size in $W$. Every chain $W\subseteq W[c_0]\subseteq W[c_1]\subseteq \cdots$ of extensions adding Cohen reals is bounded above by $W[d]$ for some $W$-generic Cohen real $d$.

This article is based upon I talk I gave at the conference on Recent Developments in Axiomatic Set Theory at the Research Institute for Mathematical Sciences (RIMS) at Kyoto University, Japan in September, 2015, and I am extremely grateful to my Japanese hosts, especially Toshimichi Usuba, for supporting my research visit there and also at the CTFM conference at Tokyo Institute of Technology just preceding it. This article includes material adapted from section section 2 of Set-theoretic geology, joint with G. Fuchs, myself and J. Reitz, and also includes a theorem that was proved in a series of conversations I had with Giorgio Venturi at the Young Set Theory Workshop 2011 in Bonn and continuing at the London 2011 summer school on set theory at Birkbeck University London.

- My talk at RIMS: Upward closure in the generic multiverse of a countable model of set theory
- My talk at CTFM: Universality and embeddability amongst the models of set theory
- G. Fuchs, J. D. Hamkins, J. Reitz, Set-theoretic geology, Annals of Pure and Applied Logic, vol. 166, iss. 4, pp. 464-501, 2015.
- Upward closure in the toy multiverse of all countable models of set theory
- Upward countable closure in the generic multiverse of forcing to add a Cohen real

What about amalgamation up to isomorphism?

Thanks for your comment! The non-amalgamation theorem was about transitive models, which are rigid, and in this case amalgamation up to isomorphism is the same thing as amalgamation, since the copy of M[c] and M[d] inside another transitive set is unique.

Ah, I see, good point. My next question would be to ask if the one could get an amalgamation up to equivalence of categories of sets. But that’s more a question for me! I’ll have a look at your paper and see if I can find a set-theoretic version of that question.

Could you say a little more about what kind of (non)amalgamation you seek? To use the Cohen-real example, we have these two models of set theory $W[c]$ and $W[d]$, which are non-amalgamable in the sense that any model of set theory containing both $c$ and $d$ will also be able to construct ordinals taller than the height of $W$ (and so this doesn’t happen in any model of set theory with the same ordinals as $W$, such as any forcing extension of $W$). We can compute the category Set in each of $W[c]$ and $W[d]$, and you are asking if these categories can be amalgamated somehow? My expectation is that the presence of $c$ and $d$ as subset objects of $\omega$ in those categories will continue to be an obstacle to amalgamation in any category-theoretic sense (unless one also has huge ordinals in the amalgamating category).

Hi Joel,

I read the paper, so now see that the caveat ‘with the same ordinals’ is key (I believe when new ordinals are in the extended model, this is called an ‘end-extension’ – is this correct? Hmm, maybe I mean, after checking Wikipedia, that it’s a combination of end-extension and regular extension….)

It should have clicked for me earlier, but now it’s obvious that category-theoretically one can ‘amalgamate’: it is just taking the pullback (over the base category Set) of the categories of sheaves on the Cohen poset. This pullback exists and is a topos, and is a sheaf topos over a particular category (I haven’t run through the standard construction that does this, but from what you say, it’s probably not just a poset or preorder). Note that this is not the pullback of the 2-valued toposes that are the categories of sets of the models W[d], W[c], but for all intents and purposes this is a minor issue.

I’m glad to hear you read my paper! I’d love to hear any comments or questions you might have about it.

Yes, as you noted it is key to insist on the same ordinals, since the models can be amalgamated without that, for example, by the model in which the non-amalgamation itself is observed. I think this is analogous to what you had observed about categories. But that amalgamating universe will not be in the same generic multiverse, since all models in the same generic multiverse have the same ordinals (a necessary but not sufficient condition).

In set-theory, one model of set theory $(M,\in^M)$ is an

end-extensionof another model $(N,\in^N)$, if it is a substructure, so that $M\subset N$ and $\in^N\upharpoonright M=\in^M$, and furthermore, $N$ adds no new elements to sets in $M$, so that $x\in^N y\in M$ implies $x\in M$. This is another way of saying that $M$ is transitive with respect to $N$. According to this definition, every forcing extension of a model is an end-extension of the model.A closely related concept is

top-extension, which means that also, the larger model $N$ does not add any new sets of rank that is an ordinal in $M$. So all the new sets of $N$ not in $M$ come on top of $M$, in terms of the von Neumann rank. For example, for any two Grothendieck universes, the taller one is a top-extension of the smaller.But one must beware a little, since many set-theorists misuse these terms, and use end-extension to mean what should be called a top-extension. So there is a little sloppiness in the literature about it.

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