The V-logic multiverse
Matteo de Ceglie (University of Salzburg) &
Claudio Ternullo (KGRC, Vienna)
In recent years, the notion of ‘set-theoretic multiverse’ has emerged and progressively gained prominence in the debate on the foundations of set theory. Several conceptions of the set-theoretic multiverse have been presented so far, all of which have advantages and disad- vantages. Hamkins’ broad multiverse ([Hamkins, 2012]), consisting of all models of all collections of set-theoretic axioms, is philosophically robust, but mathematically unattractive, as it may fail to fulfil fundamental foundational requirements of set theory. Steel’s set-generic multiverse ([Steel, 2014]) consisting of all Boolean-valued models V ???? of the axioms ZFC+Large Cardinals, is mathematically very attractive and fertile, but too restrictive. In particular, it can- not capture all possible outer models, focusing only on the set-generic extensions. Finally, Sy Friedman’s hyperuniverse conception ([Arrigoni and Friedman, 2013]), although mathematically versatile and foundationally attractive, has the main disadvantage of postulating that V is countable.
In this paper, we introduce a new conception of the set-theoretic multiverse, that is, the ‘V - logic multiverse’, which expands on mathematical work conducted within the Hyperunuverse Programme ([Antos et al, 2015], [Friedman, 2016]), but also draws on features of the set-generic multiverse, in particular, on Steel’s proposed axiomatisation of it.
V -logic is an infinitary logic (a logic admitting formulas and proofs of infinite length) whose language Lκ+,ω, in addition to symbols already used in first-order logic, consists of κ-many constants a, one for each set a ∈ V , and of a special constant symbol V , which de- notes V . In V -logic, one can ensure that the statement asserting the consistency of ZFC+ψ, for some set-theoretic statement ψ, is satisfied by some model M, if and only if M is an outer model of V . By outer model we mean here: models obtained through set-forcing, class- forcing, hyperclass-forcing and, in general, any model-theoretic technique able to produce width extensions of V . Thus, through the choice of suitable consistency statements, we can generate outer models M , endowed with specific features. The V-logic multiverse is precisely the collection of all such outer models of V .
The following observations help illustrate the adequacy of our method to produce a multiverse concept which, in our view, has better prospects than the ones mentioned above:
Contrary to the set-generic multiverse, the V-logic multiverse is broad enough to include all kinds of outer models.
Contrary to the hyperuniverse conception, the V-logic multiverse does not reduce to a collection of countable transitive models, as V does not need to be taken to be count- able.
As it stands, the V-logic multiverse may be used to pursue two fundamental research directions, both of which are ideally aimed at developing an axiomatic theory of the multiverse.
One consists in defining the V-logic multiverse of different extensions of ZFC, by taking into account such axioms as AD, PD, large cardinals, V = L and others, and investigating which relationships obtain among all such V -logic multiverses.
The second direction consists in taking V to be approximated by different structures, such as L, L-like models,Vκ, where κ is some large cardinal and investigate, for instance, whether members of the corresponding V -logic multiverses are compatible with each other, and to what extent. For instance, the L-logic multiverse maximises compatibility, but reduces the extent of structural variability among universes, thus reducing the range of alternative truth outcomes in the multiverse.
We argue that the V -logic multiverse is both mathematically more fruitful and philosophically robust than all the other multiverse conceptions, and consequently the best candidate to be the foundation of set theory and mathematics.