Kreisel’s set-theoretic problem is the problem as to whether any logical consequence of ZFC is ensured to be true. Kreisel and Boolos both proposed an answer, taking truth to mean truth in the background set-theoretic universe. This article advocates another answer, which lies at the level of models of set theory, so that truth remains the usual semantic notion. The article is divided into three parts. It first analyzes Kreisel’s set-theoretic problem and proposes one way in which any model of (...) set theory can be compared to a background universe and shown to contain internal models. It then defines logical consequence with respect to a model of ZFC, solves the model-scaled version of Kreisel’s set-theoretic problem, and presents various further results bearing on internal models. Finally, internal models are presented as accessible worlds, leading to an internal modal logic in which internal reflection corresponds to modal reflexivity, and resplendency corresponds to modal axiom 4. (shrink)

Set theory deals with the most fundamental existence questions in mathematics—questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasi-combinatorialism or combinatorial maximality. After explaining what (...) is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo—Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect. (shrink)

Deleuze (1925–1995), in the early 1980s, adopts Peirce’s (1839–1914) semiotics in order to classify the signs that the images of the cinema display. Aiming at insufflating the Peircean principles with the movement that animates the images of cinema, he provides Peirce’s triadic logic with a new category – Zeroness – which stands for the semiotic movement of cinematic images. Deleuze’s new category has impacts on the main domains of Peirce’s philosophy. Accordingly, our inquiry will focus on the irradiation of Zeroness (...) over the (a) system of categories, the (b) Peircean phenomenology (phaneroscopy), the (c) generative categorial role, the (d) semiotic effectiveness, the (e) doctrine of continuity, and the (f) logic of relatives. This article will show that for Deleuze nearly like more recent Peircean scholarship: (a) zeroth state holds a categorial status, (b) some phenomenon instantiates zeroth state, (c) zeroth state plays a generative role regarding Peirce's categories, (d) zeroth state as a phenomenological category is semiotically effective, (e) zeroth state and the doctrine of continuity, and (f) zeroth state is a “medad.” In order to assess these topics, we recover Deleuze’s advances on Peirce’s philosophy and confront them with the Peircean studies on the zeroth state. Finally, we will see that Deleuze, far from being a Peircean scholar, developed the importance of the zeroth state for Peirce’s semiotics in advance to the subsequent Peircean scholarship. Reciprocally, most of Deleuzian scholars understimate the importance of Peirce for Deleuze’s semiotics. (shrink)