author={Joel David Hamkins},
  journal={The Review of Symbolic Logic},
  pages={416 - 449}
  • J. Hamkins
  • Published 22 August 2011
  • Philosophy
  • The Review of Symbolic Logic
Abstract The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set… 
We review some conceptions of the set-theoretic multiverse and evaluate their strength. In §1, we introduce the universe/multiverse dichotomy and discuss its significance. In §2, we discuss three
Multiverse conceptions in set theory
It is argued that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory.
Multiversism and Concepts of Set: How much relativism is acceptable?
Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is
Multiverse Conceptions and the Hyperuniverse Programme December
  • Philosophy
We review different conceptions of the set-theoretic multiverse and evaluate their features and strength. In §1, we set the stage by briefly discussing the opposition between the 'universe view' and
Set Theory and Structures
This article presents a set-theoretic system that is able to talk about structures more naturally, and argues that it provides an important perspective on plausibly structural properties such as cardinality.
Abolishing Platonism in Multiverse Theories
A debated issue in the mathematical foundations in at least the last two decades is whether one can plausibly argue for the merits of treating undecidable questions of mathematics, e.g., the
An axiomatic approach to the multiverse of sets
Recent work in set theory indicates that there are many different notions of ’set’, each captured by a different collection of axioms, as proposed by J. Hamkins in [Ham11]. In this paper we strive to
What model companionship can say about the Continuum problem
. We present recent results on the model companions of set theory, placing them in the context of the current debate in the philosophy of mathematics. We start by describing the dependence of the
A multiverse perspective on the axiom of constructiblity
I shall argue that the commonly held V 6 L via maximize position, which rejects the axiom of constructibility V = L on the basis that it is restrictive, implicitly takes a stand in the pluralist
A method of interpreting extension-talk (V-logic) is presented, and it is shown how it captures satisfaction in ‘ideal’ outer models and relates to impredicative class theories.


A Natural Model of the Multiverse Axioms
This article defines that a multiverse is simply a nonempty set or class of models of ZFC set theory, allowing for a mathematized simulacrum inside V of the full philosophical multiverse (which would otherwise include universes outside V ).
TheSet-theoretic Multiverse : A IVatural Context for Set Theory
This article discusses a few emerging developments illustrating this second-order nature of set theory, and describes the principal concepts and preliminary results, mostly adapted from [4], [5] and [1], but with the main proofs omitted.
Some Second Order Set Theory
This article surveys two recent developments in set theory sharing an essential second-order nature, namely, the modal logic of forcing, oriented upward from the universe of set theory to its forcing
A simple maximality principle
This article proves that the Maximality Principle is relatively consistent with ZFC, and is equivalent to the modal theory S5.
Truth in Mathematics: The Question of Pluralism
The discovery of non-Euclidean geometries (in the nineteenth century) undermined the claim that Euclidean geometry is the one true geometry and instead led to a plurality of geometries no one of
Pointwise definable models of set theory
Every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.
Certain very large cardinals are not created in small forcing extensions
  • R. Laver
  • Mathematics
    Ann. Pure Appl. Log.
  • 2007
Internal Consistency and the Inner Model Hypothesis
This work examines Easton's strengthening of Cohen's result: Theorem 1 (Easton's Theorem), which says that a statement is internally consistent iff it holds in some inner model, under the assumption that there are inner models with large cardinals.
The ground axiom
  • J. Reitz
  • Philosophy, Economics
    Journal of Symbolic Logic
  • 2007
The Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model, is first-order expressible, and any model of ZFC has a class forcing extension which satisfies it.
The Ground Axiom asserts that the universe is not a nontrivial set-forcing extension of any inner model. Despite the apparent second-order nature of this assertion, it is first-order expressible in