The omega one of infinite chess, New York, 2012

This will be a talk on May 18, 2012 for the CUNY Set Theory Seminar.

Infinite chess is chess played on an infinite edgeless chessboard. The familiar chess pieces move about according to their usual chess rules, and each player strives to place the opposing king into checkmate.  The mate-in-$n$ problem of infinite chess is the problem of determining whether a designated player can force a win from a given finite position in at most $n$ moves.  A naive formulation of this problem leads to assertions of high arithmetic complexity with $2n$ alternating quantifiers—there is a move for white, such that for every black reply, there is a countermove for white, and so on. In such a formulation, the problem does not appear to be decidable; and one cannot expect to search an infinitely branching game tree even to finite depth. Nevertheless, in joint work with Dan Brumleve and Philipp Schlicht, confirming a conjecture of myself and C. D. A. Evans, we establish that the mate-in-$n$ problem of infinite chess is computably decidable, uniformly in the position and in $n$. Furthermore, there is a computable strategy for optimal play from such mate-in-$n$ positions. The proof proceeds by showing that the mate-in-$n$ problem is expressible in what we call the first-order structure of chess, which we prove (in the relevant fragment) is an automatic structure, whose theory is therefore decidable.  An equivalent account of the result arises from the realization that the structure of chess is interpretable in the standard model of Presburger arithmetic $\langle\mathbb{N},+\rangle$.  Unfortunately, this resolution of the mate-in-$n$ problem does not appear to settle the decidability of the more general winning-position problem, the problem of determining whether a designated player has a winning strategy from a given position, since a position may admit a winning strategy without any bound on the number of moves required. This issue is connected with transfinite game values in infinite chess, and the exact value of the omega one of chess $\omega_1^{\rm chess}$ is not known. I will also discuss recent joint work with C. D. A. Evans and W. Hugh Woodin showing that the omega one of infinite positions in three-dimensional infinite chess is true $\omega_1$: every countable ordinal is realized as the game value of such a position.

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Must there be numbers we cannot describe or define? Pointwise definability and the Math Tea argument, Bristol, April 2012

This is a talk I plan to give to the set theory seminar at the University of Bristol on April 18, 2012.

An old argument, heard at a good math tea, proceeds: “there must be some real numbers that we can neither describe nor define, since there are uncountably many reals, but only countably many definitions.” Does it withstand scrutiny? In this talk, I will discuss the phenomenon of pointwise definable models of set theory, in which every object is definable without parameters. In addition to classical and folklore results on the existence of pointwise definable models of set theory, the main new theorem is that every countable model of ZFC and indeed of GBC has an extension to a model of set theory with the same ordinals, in which every set and class is definable without parameters. This is joint work with Jonas Reitz and David Linetsky, and builds on work of S. Simpson, R. Kossak, J. Schmerl, S. Friedman and A. Enayat.

Must there be non-definable numbers? Pointwise definability and the math-tea argument, KGRC, Vienna 2011

This talk will be a part of the “Advanced Introduction” series for graduate students at the the Kurt Gödel Research Center, November 4, 2011.

An old argument, heard perhaps at math tea, proceeds: “there must be some real numbers that we can neither describe nor define, since there are uncountably many reals, but only countably many definitions.” Does it withstand scrutiny? In this talk, I will discuss the phenomenon of pointwise definable models of set theory, in which every object is definable without parameters. In addition to classical and folklore results on the existence of pointwise definable models of set theory, the main new theorem is that every countable model of ZFC and indeed of GBC has an extension to a model of set theory with the same ordinals, in which every set and class is definable without parameters. This is joint work with Jonas Reitz and David Linetsky, and builds on work of S. Simpson, R. Kossak, J. Schmerl, S. Friedman and A. Enayat.

Pointwise definable models of set theory

• J. D. Hamkins, D. Linetsky, and J. Reitz, “Pointwise definable models of set theory,” Journal of Symbolic Logic, vol. 78, iss. 1, pp. 139-156, 2013.
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One occasionally hears the argument—let us call it the math-tea argument, for perhaps it is heard at a good math tea—that there must be real numbers that we cannot describe or deﬁne, because there are are only countably many deﬁnitions, but uncountably many reals.  Does it withstand scrutiny?

This article provides an answer.  The article has a dual nature, with the first part aimed at a more general audience, and the second part providing a proof of the main theorem:  every countable model of set theory has an extension in which every set and class is definable without parameters.  The existence of these models therefore exhibit the difficulties in formalizing the math tea argument, and show that robust violations of the math tea argument can occur in virtually any set-theoretic context.

A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V=HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Godel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.

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.

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

• J. D. Hamkins, “Pointwise definable models of set theory, extended abstract,” Mathematisches Forschungsinstitut Oberwolfach Report, vol. 8, iss. 1, 02/2011, pp. 128-131, 2011.
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This is an extended abstract for the talk I gave at the Mathematisches Forschungsinstitut Oberwolfach, January 9-15, 2011.