Victoria Gitman

A Natural Model of the Multiverse Axioms

If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of (Hama). The multiverse axioms that are the focus of this article arose in connection with a continuing debate in the philosophy of set theory between the Universe view, which holds that there is a unique absolute set-theoretical universe, serving as set- theoretic background for all mathematical activity, and the Multiverse view, which holds that there are many set-theoretical worlds, each instantiating its own concept of set. We refer the reader to (Hama) and to several other articles in the same special issue of the Review of Symbolic Logic for a fuller discussion of this philosophical exchange (see also (Hamb, Ham09)). The multiverse axioms express a certain degree of richness for the set-theoretic multiverse, flowing from a perspective that denies an absolute set-theoretic background. Meanwhile, the multiverse axioms admit a purely mathematical, non-philosophical treatment, on which we shall focus here. We shall internalize the study of multi- verses to set theory by treating them as mathematical objects within ZFC, allowing for a mathematized simulacrum inside V of the full philosophical multiverse (which would otherwise include universes outside V ). Specifically, in this article we define that a multiverse is simply a nonempty set or class of models of ZFC set theory. The multiverse axioms then correspond to the features listed in Definition 1, which such a collection may or may not exhibit. Definition 1 (Multiverse Axioms). Suppose that M is a multiverse, a nonempty collection of models of ZFC. (1) The Realizability axiom holds for M if whenever M is a universe in M and N is a definable class of M satisfying ZFC from the perspective of M, then N is in M. (2) The Forcing Extension axiom holds for M if whenever M is a universe in M and P is a forcing notion in M, then M has a forcing extension of M by P, a model of the form M(G), where G is an M-generic filter for P. (3) The Class Forcing Extension axiom holds for M if whenever M is a universe in M and P is a ZFC-preserving class forcing notion in M, then M has a forcing extension of M by P, a model of the form M(G), where G is an M-generic filter for P.

Generic Vopĕnka's Principle, remarkable cardinals, and the weak Proper Forcing Axiom

We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures

Indestructibility properties of remarkable cardinals

\mathcal {C}

PROPER AND PIECEWISE PROPER FAMILIES OF REALS

C of the same type, there exist

A MODEL OF SECOND-ORDER ARITHMETIC SATISFYING AC BUT NOT DC

B\ne A

Incomparable ω1-like models of set theory

B≠A in