Tag Archives: forcing

A class of strong diamond principles

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[bibtex key=Hamkins:LaverDiamond]

In the context of large cardinals, the classical diamond principle $\Diamond_\kappa$ is easily strengthened in natural ways. When $\kappa$ is a measurable cardinal, for example, one might ask that a $\Diamond_\kappa$ sequence anticipate every subset of $\kappa$ not merely on a stationary set, but on a set of normal measure one. This is equivalent to the existence of a function $\ell:\kappa\to V_\kappa$ such that for any $A\in H(\kappa^+)$ there is an embedding $j:V\to M$ having critical point $\kappa$ with $j(\ell)(\kappa)=A$. This and similar principles formulated for many other large cardinal notions, including weakly compact, indescribable, unfoldable, Ramsey, strongly unfoldable and strongly compact cardinals, are best conceived as an expression of the Laver function concept from supercompact cardinals for these weaker large cardinal notions. The resulting Laver diamond principles can hold or fail in a variety of interesting ways.

The multiverse perspective on determinateness in set theory, Harvard, 2011

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This talk, taking place October 19, 2011, is part of the year-long Exploring the Frontiers of Incompleteness (EFI) series at Harvard University, a workshop focused on the question of determinateness in set theory, a central question in the philosophy of set theory. JDH at Harvard Streaming video will be available on-line, and each talk will be associated with an on-line discussion forum, to which links will be made here later.

In this talk, I will discuss the multiverse perspective on determinateness in set theory.  The multiverse view in set theory 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 diversity of set-theoretic possibilities, a phenomenon that challenges the universe view.  In particular, I shall argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.

Workshop materials | Article | Slides | EFI discussion forum | Video Stream

The multiverse view in set theory, Singapore 2011

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A talk at the Asian Initive for Infinity: Workshop on Infinity and Truth, July 25-29, 2011, National University of Singapore.

I shall outline and defend the Multiverse view in set theory, the view that there are many set-theoretic universes, each instantiating its own concept of set, and contrast it with the Universe view, the view that there is an absolute background set-theoretic universe.  In addition, I will discuss some recent set-theoretic developments that have been motivated and informed by a multiverse perspective, including the modal logic of forcing and the emergence of set-theoretic geology.

Slides  |  Article

Generalizations of the Kunen Inconsistency, Singapore 2011

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A talk at the Prospects of Infinity: Workshop on Set Theory  at the National University of Singapore, July 18-22, 2011.

I shall present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself, including generalizations-of-generalizations previously established by Woodin and others.  For example, there is no nontrivial elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or from V to the gHOD, or conversely from gHOD to V; indeed, there can be no nontrivial elementary embedding from any definable class to V.  Other results concern generic embeddings, definable embeddings and results not requiring the axiom of choice.  I will aim for a unified presentation that weaves together previously known unpublished or folklore results along with some new contributions.  This is joint work with Greg Kirmayer and Norman Perlmutter.

Slides | Article 

A tutorial in set-theoretic geology, London 2011

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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 satisfies 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 satisfies 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 final 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.

  Article

Slides London-2011-Set-theoretic-GeologyDownload

An introduction to Boolean ultrapowers, Bonn, 2011

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A four-lecture tutorial on the topic of Boolean ultrapowers at the Young Set Theory Workshop at the Hausdorff Center for Mathematics in Königswinter near Bonn, Germany,  March 21-25, 2011.

Boolean ultrapowers generalize the classical ultrapower construction on a power-set algebra to the context of an ultrafilter on an arbitrary complete Boolean algebra. Closely related to forcing and particularly to the use of Boolean-valued models in forcing, Boolean ultrapowers were introduced by Vopenka in order to carry out forcing as an internal ZFC construction, rather than as a meta-theoretic argument as in Cohen’s approach. An emerging interest in Boolean ultrapowers has arisen from a focus on the well-founded Boolean ultrapowers as large cardinal embeddings.

Historically, researchers have come to the Boolean ultrapower concept from two directions, from set theory and from model theory. Exemplifying the set-theoretic perspective, Bell’s classic (1985) exposition emphasizes the Boolean-valued model $V^{\mathbb{B}}$ and its quotients $V^{\mathbb{B}}/U$, rather than the Boolean ultrapower $V_U$ itself, which is not considered there. Mansfield (1970), in contrast, gives a purely algebraic, forcing-free account of the Boolean ultrapower, emphasizing its potential as a model-theoretic technique, while lacking the accompanying generic objects.

The unifying view I will explore in this tutorial is that the well-founded Boolean ultrapowers reveal the two central concerns of set-theoretic research–forcing and large cardinals–to be two aspects of a single underlying construction, the Boolean ultrapower, whose consequent close connections might be more fruitfully explored. I will provide a thorough introduction to the Boolean ultrapower construction, while assuming only an elementary graduate student-level familiarity with set theory and the classical ultrapower and forcing techniques.

Article | Abstract | Lecture Notes

Pointwise definable models of set theory, extended abstract, Oberwolfach 2011

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[bibtex key=Hamkins2011:PointwiseDefinableModelsOfSetTheoryExtendedAbstract]

This is an extended abstract for the talk I gave at the Mathematisches Forschungsinstitut Oberwolfach, January 9-15, 2011.

Slides | Main Article

 

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

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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