Satisfaction is not absolute, Dartmouth Logic Seminar, January 2014

dartmouth_campusThis will be a talk for the Dartmouth Logic Seminar on January 23rd, 2014.

Abstract. I will discuss a number of theorems showing that the satisfaction relation of first-order logic is less absolute than might have been supposed. Two models of set theory can have the same natural numbers, for example, and the same standard model of arithmetic $\langle\mathbb{N},{+},{\cdot},0,1,{\lt}\rangle$, yet disagree on their theories of arithmetic truth; two models of set theory can have the same natural numbers and a computable linear order in common, yet disagree on whether it is a well-order; two models of set theory can have the same natural numbers and the same reals, yet disagree on projective truth; two models of set theory can have a rank initial segment of the universe $\langle V_\delta,{\in}\rangle$ in common, yet disagree about whether it is a model of ZFC. The theorems are proved with elementary classical model-theoretic methods, and many of them can be considered folklore results in the subject of models of arithmetic.

Indefinite arithmetic truthOn the basis of these mathematical results, Ruizhi Yang (Fudan University, Shanghai) and I have argued that the definiteness of truth in a structure, such as with arithmetic truth in the standard model of arithmetic, cannot arise solely from the definiteness of the structure itself in which that truth resides; rather, it must be seen as a separate, higher-order ontological commitment.

Main article: Satisfaction is not absolute

Infinite chess and the theory of infinite games, Dartmouth Mathematics Colloquium, January 2014

Releasing the hordesThis will be a talk for the Dartmouth Mathematics Colloquium on January 23rd, 2014.

Dartmouth Green

Abstract. Using infinite chess as a central example—chess played on an infinite edgeless board—I shall give a general introduction to the theory of infinite games. Infinite chess is an example of what is called an open game, a potentially infinite game which when won is won at a finite stage of play, and every open game admits the theory of transfinite ordinal game values. These values provide a measure of the distance remaining to an actual victory, and when they are known, the game values provide a canonical winning strategy for the winning player. I shall exhibit

several interesting positions in infinite chess with high transfinite game values. The precise value of the omega one of chess, however, the supremum of all such ordinal game values, is an open mathematical question; in the case of infinite three-dimensional chess, meanwhile, Evans and I have proved that every countable ordinal arises as a game value. Infinite chess also illustrates an interesting engagement with computability issues. For example, there are computable infinite positions in infinite chess that are winning for white, provided that the players play according to a computable procedure of their own choosing, but which is no longer winning for white when non-computable play is allowed. Also, the mate-in-n problem for finite positions in infinite chess is computably decidable (joint work with Schlicht, Brumleve and myself), despite the high quantifier complexity of any straightforward representation of it. The talk will be generally accessible for mathematicians, particularly those with at least rudimentary knowledge of ordinals and of chess.

Poster | Slides (8mb) | Transfinite game values in infinite chess | The mate-in-n problem of infinite chess is decidable