Gabriel Uzquiano

In [12], Ernst Zermelo described a succession of models for the axioms of set theory as initial segments of a cumulative hierarchy of levelsUαVα. The recursive definition of theVα's is:Thus, a little reflection on the axioms of Zermelo-Fraenkel set theory shows thatVω, the first transfinite level of the hierarchy, is a model of all the axioms ofZFwith the exception of the axiom of infinity. And, in general, one finds that ifκis a strongly inaccessible ordinal, thenVκis a model of all of the axioms ofZF. Doubtless, when cast as a first-order theory,ZFdoes not characterize the structures 〈Vκ,∈∩〉 forκa strongly inaccessible ordinal, by the Löwenheim-Skolem theorem. Still, one of the main achievements of [12] consisted in establishing that a characterization of these models can be attained when one ventures into second-order logic. For let second-orderZFbe, as usual, the theory that results fromZFwhen the axiom schema of replacement is replaced by its second-order universal closure. Then, it is a remarkable result due to Zermelo that second-orderZFcan only be satisfied in models of the form 〈Vκ,∈∩〉 forκa strongly inaccessible ordinal.

When viewed as the most comprehensive theory of collections, set theory leaves no room for classes. But the vocabulary of classes, it is argued, provides us with compact and, sometimes, irreplaceable formulations of largecardinal hypotheses that are prominent in much very important and very interesting work in set theory. Fortunately, George Boolos has persuasively argued that plural quantification over the universe of all sets need not commit us to classes. This paper suggests that we retain the vocabulary of classes, but explain that what appears to be singular reference to classes is, in fact, covert plural reference to sets.

This paper takes a close look at the thought that mereological relations on material objects mirror, and are mirrored by, parallel mereological relations on their exact locations. This hypothesis is made more precise by means of a battery of principles from which more substantive consequences are derived. Mereological harmony turns out to entail, for example, that atomistic space is an inhospitable environment for material gunk or that Whiteheadian space is not a hospitable environment for unextended material atoms.

Philosophers often explain what could be the case in terms of what is, in fact, the case at one possible world or another. They may differ in what they take possible worlds to be or in their gloss of what is for something to be the case at a possible world. Still, they stand united by the threat of paradox. A family of paradoxes akin to the set-theoretic antinomies seem to allow one to derive a contradiction from apparently plausible principles. Some of them concern the interaction between propositions and worlds, and they appear to afford the means to map classes of propositions into propositions – or, likewise, classes of worlds into worlds – in a one-to-one fashion that leads to contradiction. Yet another family of paradoxes threaten the view that whatever could exist does, in fact, exist, which is in line with modal realism, for example. This article aims to survey and identify the source of each family of paradoxes as well as to outline some responses to them

The aim of the present paper is twofold. One task is to argue that our use of the numerical vocabulary in theory and applications determines the reference of the numerical terms more precisely than up to isomorphism. In particular our use of the numerical vocabulary in modal and counterfactual contexts of application excludes contingent existents as candidate referents for the numerical terms. The second task is to explore the impact of this conclusion on what I call semantic nominalism, which is the view that ordinary physical objects and other contingent existents are eligible to be the referents of numerical terms. I suggest that for semantic nominalism to have a chance we must reform our usage of the numerical vocabulary in modal and counterfactual contexts. I tentatively explore some ways in which this might be done and call attention to some of their costs.

The paper is concerned with the bad company problem as an instance of a more general difficulty in the philosophy of mathematics. The paper focuses on the prospects of stability as a necessary condition on acceptability. However, the conclusion of the paper is largely negative. As a solution to the bad company problem, stability would undermine the prospects of a neo-Fregean foundation for set theory, and, as a solution to the more general difficulty, it would impose an unreasonable constraint on mathematical practice.

We argue that certain modal questions raise serious problems for a modal metaphysics on which we are permitted to quantify unrestrictedly over all possibilia. In particular, we argue that, on reasonable assumptions, both David Lewis's modal realism and Timothy Williamson's necessitism are saddled with the remarkable conclusion that there is some cardinal number of the form ℵα such that there could not be more than ℵα-many angels in existence. In the last section, we make use of similar ideas to draw a moral for a recent debate in meta-ontology