The modal logic of forcing, London 2011

Posted on by
3

A talk at the conference on Set Theory and Higher-Order Logic: Foundational Issues and Mathematical Developments, following the summer school, at the University of London, Birkbeck, July 1-6, 2011.

What are the most general principles relating forceability and truth? As in Solovay’s celebrated analysis of provability, this question and its answer are naturally formulated in modal logic. Specifically, we de ne that a set theoretic assertion phi is forceable or possible if it holds in some forcing extension, and necessary if it holds in all forcing extensions. Under this forcing interpretation, the provably valid principles of forcing are exactly those in the modal theory known as S4.2. In this talk, I shall discuss this result and recent advances in this area. This is joint work with Benedikt Loewe.

London 2011 Modal Logic of ForcingDownload

See article at: The modal logic of forcing.

 

A tutorial in set-theoretic geology, London 2011

Posted on by
4

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

Research in set theory, PSC-CUNY Enhanced Research Award, 2011 – 2012

Posted on by
Reply

J. D. Hamkins, Research in set theory, PSC-CUNY Enhanced Research Award 42, 2011-2012.

Abstract.  The Principal Investigator, Professor Hamkins, pursues an active research program, with a stream of scholarly publications, international invitations to speak and distinguished grants, and is active in graduate education, currently supervising four PhD students. Professor Hamkins proposes to undertake research in the area of mathematical logic known as set theory, pursuing several projects unified by the classical Boolean ultrapower construction. Specifically, he seeks to investigate the extent to which various large cardinal extender embeddings are realized as Boolean ultrapowers, with their accompanying canonical generic objects, and to investigate the generalized Bukovsky-Dehornoy phenomenon, among other applications of the Boolean ultrapower. Professor Hamkins has a solid publication record on the topics broadly surrounding the proposed research, and has been a leading researcher on the

particular topic proposed.

Research in set theory, Simons Foundation, Collaborative Grant Award, 2011 – 2016

Posted on by
Reply

J. D. Hamkins, Research in set theory, Simons Foundation, Collaboration Grant Award, 2011-2016.

What is the theory of ZFC-Powerset? Toronto 2011

Posted on by
Reply

This was a talk at the Toronto Set Theory Seminar held April 22, 2011 at the Fields Institute in Toronto.

The theory ZFC-, consisting of the usual axioms of ZFC but with the powerset axiom removed, when axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the axiom of choice, is weaker than commonly supposed, and suffices to prove neither that a countable union of countable sets is countable, nor that $\omega_1$ is regular, nor that the Los theorem holds for ultrapowers, even for well-founded ultrapowers on a measurable cardinal, nor that the Gaifman theorem holds, that is, that every $\Sigma_1$-elementary cofinal embedding $j:M\to N$ between models of the theory is fully elementary, nor that $\Sigma_n$ sets are closed under bounded quantification. Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory obtained by using the collection axiom rather than replacement in the axiomatization above. These results extend prior work of Zarach. This is joint work with Victoria Gitman and Thomas Johnstone.

Article | Victoria Gitman’s post

An introduction to Boolean ultrapowers, Bonn, 2011

Posted on by
6

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

Posted on by
Reply

[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

Posted on by
15

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

University of Vienna, Guest Professor, June 2009

Posted on by
Reply

JosephinumI held the position of Guest Professor at the University of Vienna, Kurt Godel Research Center, in June 2009.

Research in set theory, NSF program grant, 2008- 2012

Posted on by
Reply

J. D. Hamkins, Research in Set Theory, National Science Foundation, NSF DMS 0800762, June 1, 2008 — May 31, 2012.

Summary abstract: Professor Hamkins will undertake research in the area of mathematical logic known as set theory, pursuing several projects that appear to be ripe for progress. First, the theory of models of arithmetic, usually considered to stand somewhat apart from set theory, has several fundamental questions exhibiting a deep set-theoretic nature, and an inter-speciality approach now seems called for. The most recent advances on Scott’s problem, for example, involve a sophisticated blend of techniques from models of arithmetic and the Proper Forcing Axiom.  Second, large cardinal indestructibility lies at the intersection of forcing and large cardinals, two central concerns of contemporary set-theoretic research and the core area of much of Professor Hamkins’s prior work, and recent advances have uncovered a surprisingly robust new phenomenon for relatively small large cardinals. The strongly unfoldable cardinals especially have served recently as a surprisingly efficacious substitute for supercompact cardinals in various large cardinal phenomena, including indestructibility and the consistency of fragments of the Proper Forcing Axiom. Third, Professor Hamkins will investigate questions in the emerging set-theoretic focus on second and higher order features of the set-theoretic universe.

This research in mathematical logic and set theory concentrates on topics at the foundations of mathematics, exploring the nature of mathematical infinity and the possibility of alternative mathematical universes. Our understanding of mathematical infinity, fascinating mathematicians and philosophers for centuries, has now crystallized in the large cardinal hierarchy, and a central concern of Professor Hamkins’ research will be to investigate how large cardinals are affected by forcing, the technique invented by Paul Cohen by which set theorists construct alternative mathematical universes. The diversity of these universes is astonishing, and set theorists are now able to construct models of set theory to exhibit precise pre-selected features.

In his final project, Professor Hamkins will pursue research aimed at an understanding of the most fundamental relations between the universe and these alternative mathematical worlds.

Proper forcing in large cardinal set theory, PSC-CUNY grant award, 2007 – 2008

Posted on by
Reply

J. D. Hamkins, Proper forcing in large cardinal set theory, PSC-CUNY grant award 38, 2007 – 2008.

University of Amsterdam, ILLC, Visiting Professor, 2005, 2006, 2007

Posted on by
Reply

AmsterdamWhile on sabbattical from CUNY, I held a Visiting Professor position at the Universiteit van Amsterdam at the Institute for Logic, Language and Computation during April–August 2007. During the two years previous to this, I was an NWO-sponsored Bezoekersbeurs Visiting Researcher there, during June–August 2005 and June 2006.  While in Amsterdam, I worked with Benedikt Löwe, particularly on our work concerning the modal logic of forcing.

The ground axiom, PSC-CUNY grant, 2006 – 2007

Posted on by
Reply

J. D. Hamkins, The ground axiom, PSC-CUNY grant PSC-CUNY 68198-00 37, 2006 – 2007.

Modal logics in set theory, NWO grants, 2006 – 2008

Posted on by
Reply

Modal logics in set theory, (with Benedikt Löwe), Nederlandse Organisatie voor Wetenschappelijk (B 62-619), 2006-2008.

CUNY Collaboration in Mathematical Logic, CUNY Collaboration Incentive grant, 2005 – 2007

Posted on by
Reply

CUNY Collaboration in Mathematical Logic (6 PIs), CUNY Collaboration Incentive grant, 2005 – 2007.