# Strategic thinking in infinite games, CosmoCaixa Science Museum, Barcelona, March 2023

I am deeply honored to be invited by la Caixa Foundation to give a talk in “The Greats of Science” talk series, to be held 16 March 2023 at the CosmoCaixa Science Museum in Barcelona. This talk series aspires to host “prestigious figures who have contributed towards admirable milestones, studies or discoveries,” who will bring the science to a general audience, aiming to “give viewers the chance to explore the most relevant parts of contemporary sicence through the top scientists of the moment.” Previous speakers include Jane Goodall and nearly a dozen Nobel Prize winners since 2018.

I hope to rise to those high expectations!

My topic will be: Strategic thinking in infinite games.

Have you time for an infinite game? Many familiar finite games admit natural infinitary analogues, infinite games that may captivate and challenge us with intriguing patterns and sublime complexity. Shall we have a game of infinite chess? Or how about infinite draughts, infinite Hex, infinite Wordle, or infinite Sudoku? In the Chocolatier’s game, the Chocolatier serves up an infinite stream of delicious morsels, while the Glutton aims to eat every one. These games and others illustrate the often subtle strategic aspects of infinite games, and sometimes their downright logical peculiarity. Does every infinite game admit of a winning strategy? Must optimal play be in principle computable? Let us discover the fascinating nature of infinitary strategic thinking.

The theory builds upon the classical finitary result of Zermelo (1913), the fundamental theorem of finite games, which shows that in every finite two-player game of perfect information, one of the players must have a winning strategy or both players have draw-or-better strategies. This result extends to certain infinitary games by means of the ordinal game-value analysis, which assigns transfinite ordinal values $\alpha$ to positions in a game, generalizing the familiar mate-in-$n$ idea of chess to the infinite. Current work realizes high transfinite game values in infinite chess, infinite draughts (checkers), infinite Go, and many other infinite games. The highest-known game value arising in infinite chess is the infinite ordinal $\omega^4$, and every countable ordinal arises in infinite draughts, the optimal result. Games exhibiting high transfinite ordinal game values have a surreal absurd character of play. The winning player will definitely win in finitely many moves, but the doomed losing player controls the process with absurdly long deeply nested patterns of forcing moves that must be answered, as though counting down from the infinite game value—when 0 is reached, the game is over.

# A new proof of the Barwise extension theorem, and the universal finite sequence, Barcelona Set Theory Seminar, 28 October 2020

This will be a talk for the Barcelona Set Theory Seminar, 28 October 2020 4 pm CET (3 pm UK). Contact Joan Bagaria bagaria@ub.edu for the access link.

Abstract. The Barwise extension theorem, asserting that every countable model of ZF set theory admits an end-extension to a model of ZFC+V=L, is both a technical culmination of the pioneering methods of Barwise in admissible set theory and infinitary logic and also one of those rare mathematical theorems that is saturated with philosophical significance. In this talk, I shall describe a new proof of the theorem that omits any need for infinitary logic and relies instead only on classical methods of descriptive set theory. This proof leads directly to the universal finite sequence, a Sigma_1 definable finite sequence, which can be extended arbitrarily as desired in suitable end-extensions of the universe. The result has strong consequences for the nature of set-theoretic potentialism.  This work is joint with Kameryn J. Williams.

# Every countable model of set theory is isomorphic to a submodel of its own constructible universe, Barcelona, December, 2012

This will be a talk for a set theory workshop at the University of Barcelona on December 15, 2012, organized by Joan Bagaria.

Abstract. Every countable model of set theory $M$, including every well-founded model, is isomorphic to a submodel of its own constructible universe. In other words, there is an embedding $j:M\to L^M$ that is elementary for quantifier-free assertions. The proof uses universal digraph combinatorics, including an acyclic version of the countable random digraph, which I call the countable random $\mathbb{Q}$-graded digraph, and higher analogues arising as uncountable Fraisse limits, leading to the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph, closely connected with the surreal numbers. The proof shows that $L^M$ contains a submodel that is a universal acyclic digraph of rank $\text{Ord}^M$. The method of proof also establishes that the countable models of set theory are linearly pre-ordered by embeddability: for any two countable models of set theory, one of them is isomorphic to a submodel of the other.  Indeed, the bi-embeddability classes form a well-ordered chain of length $\omega_1+1$.  Specifically, the countable well-founded models are ordered by embeddability in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory $M$ is universal for all countable well-founded binary relations of rank at most $\text{Ord}^M$; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the same proof method shows that if $M$ is any nonstandard model of PA, then every countable model of set theory—in particular, every model of ZFC—is isomorphic to a submodel of the hereditarily finite sets $HF^M$ of $M$. Indeed, $HF^M$ is universal for all countable acyclic binary relations.