Fragments of Frege's Grundgesetze and G\"odel's Constructible Universe

Frege's Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem shows that there is a model of a fragment of the Grundgesetze which defines a model of all the axioms of Zermelo-Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to G\"odel's constructible universe of sets, which G\"odel famously used to show the relative consistency of the continuum hypothesis. More specifically, our proofs appeal to Kripke and Platek's idea of the projectum within the constructible universe as well as to a weak version of uniformization (which does not involve knowledge of Jensen's fine structure theory). The axioms of the Grundgesetze are examples of abstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension. As an application, we resolve an analogue of the joint consistency problem in the predicative setting.Comment: Forthcoming in The Journal of Symbolic Logi

Doing and Showing

The persisting gap between the formal and the informal mathematics is due to an inadequate notion of mathematical theory behind the current formalization techniques. I mean the (informal) notion of axiomatic theory according to which a mathematical theory consists of a set of axioms and further theorems deduced from these axioms according to certain rules of logical inference. Thus the usual notion of axiomatic method is inadequate and needs a replacement.Comment: 54 pages, 2 figure

Nominalism In Mathematics - Modality And Naturalism

I defend modal nominalism in philosophy of mathematics - under which quantification over mathematical ontology is replaced with various modal assertions - against two sources of resistance: that modal nominalists face difficulties justifying the modal assertions that figure in their theories, and that modal nominalism is incompatible with mathematical naturalism. Shapiro argues that modal nominalists invoke primitive modal concepts and that they are thereby unable to justify the various modal assertions that figure in their theories. The platonist, meanwhile, can appeal to the set-theoretic reduction of modality, and so can justify assertions about what is logically possible through an appeal to what exists in the set-theoretic hierarchy. In chapter one, I illustrate the modal involvement of the major modal nominalist views (Chihara\u27s Constructibility Theory, Field\u27s fictionalism, and Hellman\u27s Modal Structuralism). Chapter two provides an analysis of Shapiro\u27s criticism, and a partial response to it. A response is provided in full in chapter three, in which I argue that reducing modality does not provide a means for justifying modal assertions, vitiating the accusation that modal nominalists are particularly burdened by their inability to justify modal assertions. Chapter four discusses Burgess\u27s naturalistic objection that nominalism is unscientific. I argue that Burgess\u27s naturalism is inadequately resourced to expose nominalism (modal or otherwise) as unscientific in a way that would compel a naturalist to reject nominalism. I also argue that Burgess\u27s favored moderate platonism is also guilty of being unscientific. Chapter five discusses some objections derived from Maddy\u27s naturalism, one according to which modal nominalism fails to affirm or support mathematical method, and a second according to which modal nominalism fails to be contained or accommodated by mathematical method. Though both objections serve as evidence that modal nominalism is incompatible with Maddy\u27s naturalism, I argue that Maddy\u27s naturalism is implausibly strong and that modal nominalism is compatible with forms of naturalism that relax the stronger of Maddy\u27s naturalistic principles

Against the iterative conception of set

According to the iterative conception of set, each set is a collection of sets formed prior to it. The notion of priority here plays an essential role in explanations of why contradiction-inducing sets, such as the Russell set, do not exist. Consequently, these explanations are successful only to the extent that a satisfactory priority relation is made out. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty set and raises serious epistemological concerns; but the leading realist interpretations---ontological and modal interpretations of priority---are deeply problematic as well. I conclude that the purported explanatory virtues of the iterative conception are, at present, unfounded

On arbitrary sets and ZFC

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 maximalist. 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 too fferan 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.Junta de Andalucía P07-HUM-02594Ministerio de Ciencia y Tecnología BFF2003-09579-C0