# Set-theoretic Blockchains

Joint with Miha E. Habič, Joel David Hamkins, Lukas Daniel Klausner, and Jonathan Verner.

AbstractGiven a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while preserving the nonexistence of upper bounds. We obtain several improvements of his result, using what we call the blockchain construction to build generic objects with varying degrees of mutual genericity. The method accommodates certain infinite posets, and we can realize these embeddings via a wide variety of forcing notions, while providing control over lower bounds as well. We also give a generalization to class forcing in the context of second-order set theory, and exhibit some further structure in the generic multiverse, such as the existence of exact pairs.

One position in the philosophy of set theory is the so-called multiversalist view. Under this view, there is not one ultimate conception of set, but rather many different possible universes of sets. One might liken it to the concept of space, with there being different conceptions of space, as mathematically formalized by e.g. different manifolds. This perspective suggests a diverse number of interesting mathematical questions. In this paper we take some of those questions, looking at the order structure of the generic multiverse of a model.

The generic multiverse of a universe of sets is the portion of the multiverse reachable from that universe by forcing or by going to ground models. Our paper begins with a rediscovery of some old results by Mostowski that certain posets embed into the generic multiverse, as considered as an order by inclusion. The main argument here is what we call the blockchain construction, which constructs certain Cohen reals. (Actually, Mostowski’s original context was Cohen-generic subclasses of $\mathrm{Ord}$, but the same idea works for Cohen generics in general.) The key idea is that we can control patterns of amalgamability and nonamalgamability, whether a collection of generics can be found in an outer model. We show that this phenomenon can be realized by a wide class of forcing notions. So, for instance, if one restricts to the portion of the generic multiverse generated by forcings which do not add reals, then the (non)amalgamability results still hold. We also show that the same improvements can be done in Mostowski’s original context, that of class forcing.

We then mix our metaphors, moving from blockchains to surgery. The blockchain argument builds Cohen reals from finite fragments. Whereas with surgery we can carefully graft and modify already given generics to produce generics with certain properties. This gives us a slight improvement of Mostowski’s result, getting a family of Cohen generics which can be surgically altered to witness any finite pattern of (non)amalgamability (satisfying certain obvious restrictions).

In our closing section, we look at some further order structure in the generic multiverse. We show that the exact pair phenomenon is ubiquitous in the generic multiverse. Specifically, given any countable tower of Cohen extensions and any upper bound $M[d]$ for the tower by a Cohen extension, we can find another Cohen extension $M[e]$ so that $M[d]$ and $M[e]$ form an exact pair for the tower. We close with a question: like the (non)amalgamability phenomenon, does the exact pair phenomenon generalize beyond Cohen extensions?