The Consistency of predicative fragments of frege’s grundgesetze der arithmetik

As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell?s Paradox being derivable in it.This system is, except for minor differences, full second-order logic, augmented by a single non-logical axiom, Frege?s Axiom V. It has been known for some time now that the first-order fragment of the theory is consistent. The present paper establishes that both the simple and the ramified predicative second-order fragments are consistent, and that Robinson arithmetic, Q, is relatively interpretable in the simple predicative fragment. The philosophical significance of the result is discusse

The Strength of Abstraction with Predicative Comprehension

Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege's Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic.Comment: Forthcoming in Bulletin of Symbolic Logic. Slight change in title from previous version, at request of referee

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

Predicativity, the Russell-Myhill Paradox, and Church's Intensional Logic

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church's intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms of Church's intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin's intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions.Comment: Forthcoming in The Journal of Philosophical Logi

The Julius Caesar objection

This paper argues that that Caesar problem had a technical aspect, namely, that it threatened to make it impossible to prove, in the way Frege wanted, that there are infinitely many numbers. It then offers a solution to the problem, one that shows Frege did not really need the claim that "numbers are objects", not if that claim is intended in a form that forces the Caesar problem upon us

Logicism, Ontology, and the Epistemology of Second-Order Logic

In two recent papers, Bob Hale has attempted to free second-order logic of the 'staggering existential assumptions' with which Quine famously attempted to saddle it. I argue, first, that the ontological issue is at best secondary: the crucial issue about second-order logic, at least for a neo-logicist, is epistemological. I then argue that neither Crispin Wright's attempt to characterize a `neutralist' conception of quantification that is wholly independent of existential commitment, nor Hale's attempt to characterize the second-order domain in terms of definability, can serve a neo-logicist's purposes. The problem, in both cases, is similar: neither Wright nor Hale is sufficiently sensitive to the demands that impredicativity imposes. Finally, I defend my own earlier attempt to finesse this issue, in "A Logic for Frege's Theorem", from Hale's criticisms

Hilbert's Program Then and Now

Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, "finitary" means, one should give proofs of the consistency of these axiomatic systems. Although Godel's incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert's program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.Comment: 43 page

Introduction to Abstractionism

First paragraph: Abstractionism in philosophy of mathematics has its origins in Gottlob Frege’s logicism—a position Frege developed in the late nineteenth and early twentieth century. Frege’s main aim was to reduce arithmetic and analysis to logic in order to provide a secure foundation for mathematical knowledge. As is well known, Frege’s development of logicism failed. The infamous Basic Law V— one of the six basic laws of logic Frege proposed in his magnum opus Grundgesetze der Arithmetik—is subject to Russell’s Paradox. The striking feature of Frege’s Basic Law V is that it takes the form of an abstraction principle