Blog Archives

Topic Archive: determinacy

CUNY Logic WorkshopFriday, October 2, 20152:00 pmGC 6417

Joel David Hamkins

Open determinacy for games on the ordinals is stronger than ZFC

The City University of New York

The principle of open determinacy for class games — two-player games of perfect information with plays of length ω, where the moves are chosen from a possibly proper class, such as games on the ordinals — is not provable in Zermelo-Fraenkel set theory ZFC or Gödel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for first-order truth, and indeed a transfinite tower of truth predicates TR_α for iterated truth-about-truth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of elementary transfinite recursion ETR over well-founded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC + Pi^1_1 comprehension, a proper fragment of Kelley-Morse set theory KM.

This is joint work with Victoria Gitman, with helpful participation of Thomas Johnstone.

See also related article: V. Gitman, J.D. Hamkins, Open determinacy for class games, submitted.

For further information and commentary concerning this talk, please see the related post on my blog.

Set theory seminarMonday, March 30, 20155:00 pmGC 3309

Ralf Schindler

Harrington’s Principle and remarkable cardinals

University of Münster

It is open whether $\Pi^1_1$ determinacy implies the existence of $0^{\#}$ in 3rd order arithmetic, call it $Z_3$. We compute the large cardinal strength of $Z_3$ plus “there is a real $x$ such that every $x$-admissible is an $L$-cardinal.” This is joint work with Yong Cheng.

Scott Cramer
Rutgers University
Scott Cramer received his Ph.D. from the University of California-Berkeley in 2012, with a thesis on reflection of large cardinals, written under the supervision of Hugh Woodin. He is presently a Triennial Assistant Professor at Rutgers University, working in set theory, with specific interests in very large cardinals and determinacy axioms.
Tuesday, April 30, 20134:00 pmYeshiva UniversityFurst Hall, Amsterdam Ave. & 185th Street.

Joel David Hamkins

The theory of infinite games, with examples, including infinite chess

The City University of New York

This will be a talk on April 30, 2013 for a joint meeting of the Yeshiva University Mathematics Club and the Yeshiva University Philosophy Club. I will give a general introduction to the theory of infinite games, suitable for mathematicians and philosophers. What does it mean to play an infinitely long game? What does it mean to have a winning strategy for such a game? Is there any reason to think that every game should have a winning strategy for one player or another? Could there be a game, such that neither player has a way to force a win? Must every computable game have a computable winning strategy? I will present several game paradoxes and example infinitary games, including an infinitary version of the game of Nim, and several examples from infinite chess.

CUNY Logic WorkshopFriday, March 22, 20134:00 pmGC 6417Note special time

Philip Welch

Determinacy in analysis and beyond

University of Bristol

Recently Montalban and Shore derived precise limits to the amount of determinacy provable in second order arithmetic.  We review some of the results in this area and recent work on lifting this to a setting of ZF^- with a single measurable cardinal.