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We prove a number of results on the determinacy of $sigma$-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable unions, and projections. We first prove the equivalence between $sigma$-projective determinacy and the determinacy of certain classes of games of variable length ${<}omega^2$ (Theorem 2.4). We then give an elementary proof of the determinacy of $sigma$-projective sets from optimal large-cardinal hypotheses (Theorem 4.4). Finally, we show how to generalize the proof to obtain proofs of the determinacy of $sigma$-projective games of a given countable length and of games with payoff in the smallest $sigma$-algebra containing the projective sets, from corresponding assumptions (Theorems 5.1 and 5.4).
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The notion of a textbf{$boldsymbol{mathcal{C}}$-filtered} object, where $mathcal{C}$ is some (typically small) collection of objects in a Grothendieck category, has become ubiquitous since the solution of the Flat Cover Conjecture around the year 2000. We introduce the textbf{$boldsymbol{mathcal{C}}$-Filtration Game of length $boldsymbol{omega_1}$} on a module, paying particular attention to the case where $mathcal{C}$ is the collection of all countably presented, projective modules. We prove that Martins Maximum implies the determinacy of many $mathcal{C}$-Filtration Games of length $omega_1$, which in turn imply the determinacy of certain Ehrenfeucht-Fraiss{e} games of length $omega_1$; this allows a significant strengthening of a theorem of Mekler-Shelah-Vaananen cite{MR1191613}. Also, Martins Maximum implies that if $R$ is a countable hereditary ring, the class of textbf{$boldsymbol{sigma}$-closed potentially projective modules}---i.e., those modules that are projective in some $sigma$-closed forcing extension of the universe---is closed under $<aleph_2$-directed limits. We also give an example of a (ZFC-definable) class of abelian groups that, under the ordinary subgroup relation, constitutes an Abstract Elementary Class (AEC) with Lowenheim-Skolem number $aleph_1$ in some models in set theory, but fails to be an AEC in other models of set theory.
Let $M^sharp_n(mathbb{R})$ denote the minimal active iterable extender model which has $n$ Woodin cardinals and contains all reals, if it exists, in which case we denote by $M_n(mathbb{R})$ the class-sized model obtained by iterating the topmost measure of $M_n(mathbb{R})$ class-many times. We characterize the sets of reals which are $Sigma_1$-definable from $mathbb{R}$ over $M_n(mathbb{R})$, under the assumption that projective games on reals are determined: (1) for even $n$, $Sigma_1^{M_n(mathbb{R})} = Game^mathbb{R}Pi^1_{n+1}$; (2) for odd $n$, $Sigma_1^{M_n(mathbb{R})} = Game^mathbb{R}Sigma^1_{n+1}$. This generalizes a theorem of Martin and Steel for $L(mathbb{R})$, i.e., the case $n=0$. As consequences of the proof, we see that determinacy of all projective games with moves in $mathbb{R}$ is equivalent to the statement that $M^sharp_n(mathbb{R})$ exists for all $ninmathbb{N}$, and that determinacy of all projective games of length $omega^2$ with moves in $mathbb{N}$ is equivalent to the statement that $M^sharp_n(mathbb{R})$ exists and satisfies $mathsf{AD}$ for all $ninmathbb{N}$.
We determine the consistency strength of determinacy for projective games of length $omega^2$. Our main theorem is that $boldsymbolPi^1_{n+1}$-determinacy for games of length $omega^2$ implies the existence of a model of set theory with $omega + n$ Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals $A$ such that $M_n(A)$, the canonical inner model for $n$ Woodin cardinals constructed over $A$, satisfies $A = mathbb{R}$ and the Axiom of Determinacy. Then we argue how to obtain a model with $omega + n$ Woodin cardinal from this. We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length $omega^2$ with payoff in $Game^mathbb{R} boldsymbolPi^1_1$ or with $sigma$-projective payoff.
In 2011, Rideau and Winskel introduced concurrent games and strategies as event structures, generalizing prior work on causal formulations of games. In this paper we give a detailed, self-contained and slightly-updated account of the results of Rideau and Winskel: a notion of pre-strategy based on event structures; a characterisation of those pre-strategies (deemed strategies) which are preserved by composition with a copycat strategy; and the construction of a bicategory of these strategies. Furthermore, we prove that the corresponding category has a compact closed structure, and hence forms the basis for the semantics of concurrent higher-order computation.
It is shown, from hypotheses in the region of $omega^2$ Woodin cardinals, that there is a transitive model of KP + AD$_mathbb{R}$ containing all reals.
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