We isolate two abstract determinacy theorems for games of length $\omega_1$ from work of Neeman and use them to conclude, from large-cardinal assumptions and an iterability hypothesis in the region of measurable Woodin cardinals thatif the Continuum Hypothesis holds, then all games of length $\omega_1$ which are provably $\Delta_1$ -definable from a universally Baire parameter are determined;all games of length $\omega_1$ with payoff constructible relative to the play are determined; andif the Continuum Hypothesis holds, then there is a model of (...) ${\mathsf{ZFC}}$ containing all reals in which all games of length $\omega_1$ definable from real and ordinal parameters are determined. (shrink)

Takeuti introduced an infinitary proof system for determinate logic and showed that for transitive models of Zermelo-Fraenkel set theory with the Axiom of Dependent Choice that contain all reals, the cut-elimination theorem is equivalent to the Axiom of Determinacy, and in particular contradicts the Axiom of Choice. We consider variants of Takeuti's theorem without assuming the failure of the Axiom of Choice. For instance, we show that if one removes atomic formulae of infinite arity from the language of Takeuti's proof (...) system, then cut elimination is equivalent to a determinacy hypothesis provable, e.g., in ZFC + “there are infinitely many Woodin cardinals.” A slight extension of the proof system admits cut elimination for countable sequents under the same assumptions. A simple modification of the proof system yields analogs of the results above for the Axiom of Real Determinacy and uncountably many Woodin cardinals. (shrink)

It is shown that the determinacy of $G_{\delta \sigma }$ games of length $\omega ^2$ is equivalent to the existence of a transitive model of ${\mathsf {KP}} + {\mathsf {AD}} + \Pi _1\textrm {-MI}_{\mathbb {R}}$ containing $\mathbb {R}$. Here, $\Pi _1\textrm {-MI}_{\mathbb {R}}$ is the axiom asserting that every monotone $\Pi _1$ operator on the real numbers has an inductive fixpoint.

We characterize the determinacy of $F_\sigma $ games of length $\omega ^2$ in terms of determinacy assertions for short games. Specifically, we show that $F_\sigma $ games of length $\omega ^2$ are determined if, and only if, there is a transitive model of ${\mathsf {KP}}+{\mathsf {AD}}$ containing $\mathbb {R}$ and reflecting $\Pi _1$ facts about the next admissible set.As a consequence, one obtains that, over the base theory ${\mathsf {KP}} + {\mathsf {DC}} + ``\mathbb {R}$ exists,” determinacy for $F_\sigma $ (...) games of length $\omega ^2$ is stronger than ${\mathsf {AD}}$, but weaker than ${\mathsf {AD}} + \Sigma _1$ -separation. (shrink)

This article describes new aspects of hysteresis dynamics which have been uncovered through computer experiments. There are several motivations to be interested in fast-slow dynamics. For instance, many physiological or biological systems display different time scales. The bursting oscillations which can be observed in neurons, β-cells of the pancreas and population dynamics are essentially studied via bifurcation theory and analysis of fast-slow systems (Keener and Sneyd, 1998; Rinzel, 1987). Hysteresis is a possible mechanism to generate bursting oscillations. A first part (...) of this article presents the computer techniques (the dotted-phase portrait, the bifurcation of the fast dynamics and the wave form) we have used to represent several patterns specific to hysteresis dynamics. This framework yields a natural generalization to the notion of bursting oscillations where, for instance, the active phase is chaotic and alternates with a quiescent phase. In a second part of the article, we emphasize the evolution to chaos which is often associated with bursting oscillations on the specific example of the Hindmarsh–Rose system. This evolution to chaos has already been studied with classical tools of dynamical systems but we give here numerical evidence on hysteresis dynamics and on some aspects of the wave form. The analytical proofs will be given elsewhere. (shrink)

In their thought-provoking article, Sedlakova and Trachsel (2023) defend the view that the status—both epistemic and ethical—of Conversational Artificial Intelligence (CAI) used in psychotherapy is complicated. While therapeutic CAI seems to be more than a mere tool implementing particular therapeutic techniques, it falls short of being a “digital therapist.” One of the main arguments supporting the latter claim is that even though “the interaction with CAI happens in the course of conversation… the conversation is profoundly different from a conversation with (...) a human therapist” (Sedlakova and Trachsel 2023, 8). In particular, unlike a human therapist, CAI cannot help its users gain new insight and self-understanding (Sedlakova and Trachsel 2023). We agree that currently available therapeutic CAI cannot be considered a “digital therapist,” however, we think that the issue surrounding the acquisition of new self-understanding in the interaction with therapeutic CAI is more complicated than Sedlakova and Trachsel suggest. (shrink)

The use of tensed language and the metaphor of set ‘formation’ found in informal descriptions of the iterative conception of set are seldom taken at all seriously. Both are eliminated in the nonmodal stage theories that formalise this account. To avoid the paradoxes, such accounts deny the Maximality thesis, the compelling thesis that any sets can form a set. This paper seeks to save the Maximality thesis by taking the tense more seriously than has been customary (although not literally). A (...) modal stage theory, MST, is developed in a bimodal language, governed by a tenselike logic. Such a language permits a very natural axiomatisation of the iterative conception, which upholds the Maximality thesis. It is argued that the modal approach is consonant with mathematical practice and a plausible metaphysics of sets and shown that MST interprets a natural extension of Zermelo set theory less the axiom of Infinity and, when extended with a further axiom concerning the extent of the hierarchy, interprets Zermelo–Fraenkel set theory. (shrink)

Growing demand for broadly accessible mental health care, together with the rapid development of new technologies, trigger discussions about the feasibility of psychotherapeutic interventions based on interactions with Conversational Artificial Intelligence (CAI). Many authors argue that while currently available CAI can be a useful supplement for human-delivered psychotherapy, it is not yet capable of delivering fully fledged psychotherapy on its own. The goal of this paper is to investigate what are the most important obstacles on our way to developing CAI (...) systems capable of delivering psychotherapy in the future. To this end, we formulate and discuss three challenges central to this quest. Firstly, we might not be able to develop effective AI-based psychotherapy unless we deepen our understanding of what makes human-delivered psychotherapy effective. Secondly, assuming that it requires building a therapeutic relationship, it is not clear whether psychotherapy can be delivered by non-human agents. Thirdly, conducting psychotherapy might be a problem too complicated for narrow AI, i.e., AI proficient in dealing with only relatively simple and well-delineated tasks. If this is the case, we should not expect CAI to be capable of delivering fully-fledged psychotherapy until the so-called “general” or “human-like” AI is developed. While we believe that all these challenges can ultimately be overcome, we think that being mindful of them is crucial to ensure well-balanced and steady progress on our path to AI-based psychotherapy. (shrink)