Browse Papers

Using an axiomatization of second-order arithmetic (essentially second-order Peano Arithmetic without the Successor Axiom), arithmetic's basic operations are defined and its fundamental laws, up to unique prime factorization, are proven. Two manners of expressing a system's consistency are presented - the "Godel" consistency, where a wff is represented by a natural number, and the "real" consistency, where a wff is represented as a second-order sequence, which is a stronger notion. It is shown that the system can prove at least its Godel consistency and that closely allied systems can prove their real consistency.

In this paper we outline and discuss various solutions to a restricted, but we think, more interesting version of the infamous Caesar Problem. This restricted version, labelled C-R Problem, occurs in contexts where we have two distinct abstraction principles:

AP_{@_{1}} (\forall X)(\forall Y)[@_{1}(X)=@_{1}(Y)\leftrightarrow E_{@_{1}}(X,Y)]

AP_{@_{2}} (\forall X)(\forall Y)[@_{2}(X)=@_{2}(Y)\leftrightarrow E_{@_{2}}(X,Y)]

and want to settle cross-sortal identity claims of the form:

@_{1}(P)=@_{2}(Q)

Both abstraction principles, however, are silent with regard to this identity -- a special instance of the Caesar Problem. In what follows, we outline two distinct strategies to resolve the C-R problem. The first strategy decides such cross-abstraction identities in terms of whether or not the equivalence relations appearing on the right hand side of the abstraction principles are identical, while the second strategy settles such identities by appeal to the relevant equivalence classes. We then focus our discussion on the latter approach and offer three ways of implementing this strategy. Ultimately, we argue that this strategy fails, as each attempt to appeal to equivalence classes faces unsurmountable difficulties.

Second-order Peano Arithmetic minus the Successor Axiom is developed from first principles through Quadratic Reciprocity and a proof of self-consistency. This paper combines 4 other papers of the author in a self-contained exposition.

We extend an independence result proved in our earlier paper "Solovay's Theorem Cannot Be Simplified" (Annals of Pure and Applied Logic 112 (2001)). Our method uses the Barwise–Kreisel Compactness Theorem.

Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's principle. Husserl, and later Parsons, objected that there is no such close connection that our most primitive conception of cardinality arises from our grasp of the practice of counting. I argue, however, that Frege was close to right, that our concept of cardinal number is closely connected with a notion like that of one-one correspondence, a more primitive notion we might call just as many.

This paper discusses Frege's account of definition by induction in Grundgesetze and the two key theorems Frege proves using it.

Co-authored discussion note on Kit Fine's "Limits of Abstraction" with special focus on the more philosophical aspects of the book (chapter one and two).