This thesis presents my contributions to various aspects of the theory of universally Baire sets. One of these aspects is the smallest inner model containing all reals whose all sets of reals are universally Baire (viz., $L(\mathbb {R})$) and its relation to its inner model $\mathsf {HOD}$. We verify here that $\mathsf {HOD}^{L(\mathbb {R})}$ enjoys a form of local definability inside $L(\mathbb {R})$, further justifying its characterization as a “core model” in $L(\mathbb {R})$. We then study a “bottom-up” construction of more complicated universally Baire sets (more generally, determined sets). This construction allows us to give an “L-like” description of the minimum model of $\mathsf {AD}_{\mathbb {R}} + \mathsf {Cof}(\Theta ) = \Theta $. A consequence of this description is that this minimum model is contained in the Chang-plus model. Our construction, together with Woodin’s work on the Chang-plus model, shows that a proper class of Woodin cardinals which are limits of Woodin cardinals implies the existence of a hod mouse with a measurable limit of Woodin cardinals whose strategy is universally Baire.

Another aspect of the theory of universally Baire sets is the generic absoluteness and maximality associated with them. We include some results concerning generic $\Sigma _1^{H(\omega _2)}$-absoluteness with universally Baire sets as predicates or parameters, as well as generic $\Pi _2^{H(\omega _2)}$-maximality with universally Baire sets as predicates. In the second case, we are led to consider the general question of when a model of an infinitary propositional formula can be added by a stationary-set-preserving poset. We characterize when this happens in terms of a game which is a variant of the Model Existence Game. We then give a sufficient condition for this in terms of generic embeddings.

Abstract taken directly from the thesis

E-mail: obrad@math.ucla.edu

URL: https://hal.science/view/index/docid/5273407

Within the determinacy setting, ${\mathscr {P}({\omega _1})}$ is regular (in the sense of cofinality) with respect to many known cardinalities and thus there is substantial evidence to support the conjecture that ${\mathscr {P}({\omega _1})}$ has globally regular cardinality. However, there is no known information about the regularity of ${\mathscr {P}(\omega _2)}$. It is not known if ${\mathscr {P}(\omega _2)}$ is even $2$-regular under any determinacy assumptions. The article will provide the following evidence that ${\mathscr {P}(\omega _2)}$ may possibly be ${\omega _1}$-regular: Assume $\mathsf {AD}^+$. If $\langle A_\alpha : \alpha < {\omega _1} \rangle $ is such that ${\mathscr {P}(\omega _2)} = \bigcup _{\alpha < {\omega _1}} A_\alpha $, then there is an $\alpha < {\omega _1}$ so that $\neg (|A_\alpha | \leq |[\omega _2]^{<\omega _2}|)$.

We prove that every $\Sigma ^0_2$ Gale-Stewart game can be won via a winning strategy $\tau $ which is $\Delta _1$-definable over $L_{\delta }$, the $\delta $th stage of Gödel’s constructible universe, where $\delta = \delta _{\sigma ^1_1}$, strengthening a theorem of Solovay from the 1970s. Moreover, the bound is sharp in the sense that there is a $\Sigma ^0_2$ game with no strategy $\tau $ which is witnessed to be winning by an element of $L_{\delta }$.

Carnap’s (Categoricity) Problem concerns the relationship between (rules of) inference and model-theoretic values. In particular, it asks whether proof-theoretic constraints are ‘strong enough’ to uniquely determine intended semantic values. Carnap [20] demonstrated that already in the classical bivalent setting this is not the case for the majority of the usual logical constants. To remedy this underdetermination of ‘semantics by syntax’ a variety of solution strategies has been explored in the literature. This article is a philosophical-logical survey of these attempts, comparing them with respect to scope, motivation, and success. Besides the mathematical interest held by Carnap’s Problem, the underdetermination it uncovers has significant consequences for a variety of philosophical projects and positions, warranting a systematic study of attempts at resolving it.

This paper explores the role of the cost channel in a behavioral New Keynesian model where households and firms have different degrees of cognitive discounting. Our findings are summarized as follows. First, we demonstrate how the degree of cognitive discounting significantly affects the determinacy condition through the cost channel model. Second, a high degree of cognitive discounting attenuates the response of inflation to a monetary tightening shock, and the cost channel amplifies this effect. Third, the degree of cognitive discounting significantly impacts the effect of the cost channel on the design of optimal monetary policy.

We provide new evidence about US monetary policy using a model that: (i) estimates time-varying monetary policy weights without relying on stylized theoretical assumptions; (ii) allows for endogenous breakdowns in the relationship between interest rates, inflation, and output; and (iii) generates a unique measure of monetary policy activism that accounts for economic instability. The joint incorporation of endogenous time-varying uncertainty about the monetary policy parameters and the stability of the relationship between interest rates, inflation, and output materially reduces the probability of determinate monetary policy. The average probability of determinacy over the period post-1982 to 1997 is below 60% (hence well below seminal estimates of determinacy probabilities that are close to unity). Post-1990, the average probability of determinacy is 75%, falling to approximately 60% when we allow for typical levels of trend inflation.

A short core model induction proof of $\mathsf {AD}^{L(\mathbb {R})}$ from $\mathsf {TD} + \mathsf {DC}_{\mathbb {R}}$.

Work by Chomsky et al. (2019) and Epstein et al. (2018) develops a third-factor principle of computational efficiency called “Determinacy”, which rules out “ambiguous” syntactic rule-applications by requiring one-to-one correspondences between the input or output of a rule and a single term in the domain of that rule. This article first adopts the concept of “Input Determinacy” articulated by Goto and Ishii (2019, 2020), who apply Determinacy specifically to the input of operations like Merge, and then proposes to extend Determinacy to the labeling procedure developed by Chomsky (2013, 2015). In particular, Input Determinacy can explain restrictions on labeling in contexts where multiple potential labels are available (labeling ambiguity), and it can also provide an explanation for Chomsky's (2013, 2015) proposal that syntactic movement of an item (“Internal Merge”) renders that item invisible to the labeling procedure.

Assume $\mathsf {ZF} + \mathsf {AD}$ and all sets of reals are Suslin. Let $\Gamma $ be a pointclass closed under $\wedge $, $\vee $, $\forall ^{\mathbb {R}}$, continuous substitution, and has the scale property. Let $\kappa = \delta (\Gamma )$ be the supremum of the length of prewellorderings on $\mathbb {R}$ which belong to $\Delta = \Gamma \cap \check \Gamma $. Let $\mathsf {club}$ denote the collection of club subsets of $\kappa $. Then the countable length everywhere club uniformization holds for $\kappa $: For every relation $R \subseteq {}^{<{\omega _1}}\kappa \times \mathsf {club}$ with the property that for all $\ell \in {}^{<{\omega _1}}\kappa $ and clubs $C \subseteq D \subseteq \kappa $, $R(\ell ,D)$ implies $R(\ell ,C)$, there is a uniformization function $\Lambda : \mathrm {dom}(R) \rightarrow \mathsf {club}$ with the property that for all $\ell \in \mathrm {dom}(R)$, $R(\ell ,\Lambda (\ell ))$. In particular, under these assumptions, for all $n \in \omega $, $\boldsymbol {\delta }^1_{2n + 1}$ satisfies the countable length everywhere club uniformization.

Schmidt’s game and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games, $\mathsf {AD}_{\mathbb R}$, which is a much stronger axiom than that asserting all integer games are determined, $\mathsf {AD}$. One of our main results is a general theorem which under the hypothesis $\mathsf {AD}$ implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt’s $(\alpha ,\beta ,\rho )$ game on $\mathbb R$ is determined from $\mathsf {AD}$ alone, but on $\mathbb R^n$ for $n \geq 3$ we show that $\mathsf {AD}$ does not imply the determinacy of this game. We then give an application of simple strategies and prove that the winning player in Schmidt’s $(\alpha , \beta , \rho )$ game on $\mathbb {R}$ has a winning positional strategy, without appealing to the axiom of choice. We also prove several other results specifically related to the determinacy of Schmidt’s game. These results highlight the obstacles in obtaining the determinacy of Schmidt’s game from $\mathsf {AD}$.

We consider a real-valued function f defined on the set of infinite branches X of a countably branching pruned tree T. The function f is said to be a limsup function if there is a function $u \colon T \to \mathbb {R}$ such that $f(x) = \limsup _{t \to \infty } u(x_{0},\dots ,x_{t})$ for each $x \in X$. We study a game characterization of limsup functions, as well as a novel game characterization of functions of Baire class 1.

We show that, assuming the Axiom of Determinacy, every non-selfdual Wadge class can be constructed by starting with those of level $\omega _1$ (that is, the ones that are closed under Borel preimages) and iteratively applying the operations of expansion and separated differences. The proof is essentially due to Louveau, and it yields at the same time a new proof of a theorem of Van Wesep (namely, that every non-selfdual Wadge class can be expressed as the result of a Hausdorff operation applied to the open sets). The exposition is self-contained, except for facts from classical descriptive set theory.