# Some second order set theory

• J. D. Hamkins, “Some second order set theory,” in Logic and its applications, R.~Ramanujam and S.~Sarukkai, Eds., Springer, 2009, vol. 5378, pp. 36-50.
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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 extensions; and set-theoretic geology, oriented downward from the universe to the inner models over which it arises by forcing. The research is a mixture of ideas from several parts of logic, including, of course, set theory and forcing, but also modal logic, finite combinatorics and the philosophy of mathematics, for it invites a mathematical engagement with various philosophical views on the nature of mathematical existence.

# The ground axiom is consistent with $V\ne{\rm HOD}$

• J. D. Hamkins, J. Reitz, and W. Woodin, “The ground axiom is consistent with $V\ne{\rm HOD}$,” Proc.~Amer.~Math.~Soc., vol. 136, iss. 8, pp. 2943-2949, 2008.
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Abstract. 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 set theory. The previously known models of the Ground Axiom all satisfy strong forms of $V=\text{HOD}$. In this article, we show that the Ground Axiom is relatively consistent with $V\neq\text{HOD}$. In fact, every model of ZFC has a class-forcing extension that is a model of $\text{ZFC}+\text{GA}+V\neq\text{HOD}$. The method accommodates large cardinals: every model of ZFC with a supercompact cardinal, for example, has a class-forcing extension with $\text{ZFC}+\text{GA}+V\neq\text{HOD}$ in which this supercompact cardinal is preserved.

# The Ground Axiom

• J. D. Hamkins, “The Ground Axiom,” Mathematisches Forschungsinstitut Oberwolfach Report, vol. 55, pp. 3160-3162, 2005.
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This is an extended abstract for a talk I gave at the 2005 Workshop in Set Theory at the Mathematisches Forschungsinstitut Oberwolfach.

# A tutorial in set-theoretic geology, London 2011

A three-lecture mini-course tutorial in set-theoretic geology at the summer school Set Theory and Higher-Order Logic: Foundational Issues and Mathematical Developments, August 1-6, 2011, University of London, Birkbeck.

The technique of forcing in set theory is customarily thought of as a method for constructing outer as opposed to inner models of set theory; one starts in a ground model V and considers the possible forcing extensions V[G] of it. A simple switch in perspective, however, allows us to use forcing to describe inner models, by considering how a given universe V may itself have arisen by forcing. This change in viewpoint leads to the topic of set-theoretic geology, aiming to investigate the structure and properties of the ground models of the universe. In this tutorial, I shall present some of the most interesting initial results in the topic, along with an abundance of open questions, many of which concern fundamental issues.

A ground of the universe V is an inner model W of ZFC over which the universe V=W[G] is a forcing extension. The model V satis es the Ground Axiom of there are no such W properly contained in V. The model W is a bedrock of V if it is a ground of V and satis es the Ground Axiom. The mantle of V is the intersection of all grounds of V, and the generic mantle is the intersection of all grounds of all set-forcing extensions. The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC, and this can be proved while also controlling the HOD of the nal model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to the outer core, what remains when all outer layers of forcing have been stripped away. Many fundamental questions remain open.

# The set-theoretic multiverse: a model-theoretic philosophy of set theory, Paris, 2010

A talk at the Philosophy and Model Theory conference held June 2-5, 2010 at the Université Paris Ouest Nanterre.

Set theorists commonly regard set theory as an ontological foundation for the rest of mathematics, in the sense that other abstract mathematical objects can be construed fundamentally as sets, enjoying a real mathematical existence as sets accumulate to form the universe of all sets. The Universe view—perhaps it is the orthodox view among set theorists—takes this universe of sets to be unique, and holds that a principal task of set theory is to discover its fundamental truths. For example, on this view, interesting set-theoretical questions, such as the Continuum Hypothesis, will have definitive final answers in this universe. Proponents of this view point to the increasingly stable body of regularity features flowing from the large cardinal hierarchy as indicating in broad strokes that we are on the right track towards these final answers.

A paradox for the orthodox view, however, is the fact that the most powerful tools in set theory are most naturally understood as methods for constructing alternative set-theoretic universes. With forcing and other methods, we seem to glimpse into alternative mathematical worlds, and are led to consider a model-theoretic, multiverse philosophical position. In this talk, I shall describe and defend the Multiverse view, which takes these other worlds at face value, holding that there are many set-theoretical universes. This is a realist position, granting these universes a full mathematical existence and exploring their interactions. The multiverse view remains Platonist, but it is second-order Platonism, that is, Platonism about universes. I shall argue that set theory is now mature enough to fruitfully adopt and analyze this view. I shall propose a number of multiverse axioms, provide a multiverse consistency proof, and describe some recent results in set theory that illustrate the multiverse perspective, while engaging pleasantly with various philosophical views on the nature of mathematical existence.

Slides  | Article | see related Singapore talk