Frege's logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the "neologicist" approach of Hale and Wright. Less attention has been given to Frege's extensionalist program, according to which arithmetic is to be reconstructed in terms of a theory of extensions of concepts. This paper deals just with such a theory. We present a system of second-order logic augmented with a predicate representing the fact that an object x is the extension of a concept C, together with extra-logical axioms governing such a predicate, and show that arithmetic can be obtained in such a framework. As a philosophical payoff, we investigate the status of the so-called Hume's Principle and its connections to the root of the contradiction in Frege's system.

We study connections between asymptotic structure in a Banach space and model theoretic properties of the space. We show that, in an asymptotic sense, a sequence $(x_n)$ in a Banach space X generates copies of one of the classical sequence spaces $\ell_p$ or $c_0$ inside X (almost isometrically) if and only if the quantifier-free types approximated by $(x_n)$ inside X are quantifier-free definable. More precisely, if $(x_n)$ is a bounded sequence X such that no normalized sequence of blocks of $(x_n)$ converges, then the following two conditions are equivalent. (1) There exists a sequence $(y_n)$ of blocks of $(x_n)$ such that for every finite dimensional subspace E of X, every quantifier-free type over $E +\overline{\rm span}\{y_n\mid n\in \mathbb{N}\}$ is quantifier-free definable. (2) One of the following two conditions holds: (a) there exists $1\le p< \infty$ such that for every $\epsilon>0$ and every finite dimensional subspace E of X there exists a sequence of blocks of $(x_n)$ which is $(1+\epsilon)$equivalent over E to the standard unit basis of $\ell_p$; (b) for every $\epsilon>0$ and every finite dimensional subspace E of X there exists a sequence of blocks of $(x_n)$ which is $(1+\epsilon)$-equivalent over E to the standard unit basis of $c_0$. Several byproducts of the proof are analyzed.

We define a program size complexity function $H^\infty$ as a variant of the prefix-free Kolmogorov complexity, based on Turing monotone machines performing possibly unending computations. We consider definitions of randomness and triviality for sequences in ${\{0,1\}}^\omega$ relative to the $H^\infty$ complexity. We prove that the classes of Martin-Löf random sequences and $H^\infty$-random sequences coincide and that the $H^\infty$-trivial sequences are exactly the recursive ones. We also study some properties of $H^\infty$ and compare it with other complexity functions. In particular, $H^\infty$ is different from $H^A$, the prefix-free complexity of monotone machines with oracle A.

Visser's rules form a basis for the admissible rules of ${\sf IPC}$. Here we show that this result can be generalized to arbitrary intermediate logics: Visser's rules form a basis for the admissible rules of any intermediate logic ${\sf L}$ for which they are admissible. This implies that if Visser's rules are derivable for ${\sf L}$ then ${\sf L}$ has no nonderivable admissible rules. We also provide a necessary and sufficient condition for the admissibility of Visser's rules. We apply these results to some specific intermediate logics and obtain that Visser's rules form a basis for the admissible rules of, for example, De Morgan logic, and that Dummett's logic and the propositional Gödel logics do not have nonderivable admissible rules.

Let $\mathsf{I\Sigma_1}$ be the fragment of elementary Peano arithmetic in which induction is restricted to $\Sigma_1$-formulas. More than three decades ago, Parsons showed that the provably total functions of $\mathsf{I\Sigma_1}$ are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the $\exists \forall \exists$-consequences of universal theories. We give a self-contained proof requiring only basic knowledge of mathematical logic.

The expressive truth functions of two-valued logic have all been identified. This paper begins the task of identifying the expressive truth functions of n-valued logic by characterizing the unary ones. These functions have distinctive algebraic, semantic, and closure-theoretic properties.