The well known Wiener-Kuratowski explicit definition of the ordered pair, which sets ⟨x, y⟩ = {{x}, {x, y}}, works well in many set theories but fails for those with classes which cannot be members of singletons. With the aid of the Axiom of Foundation, we propose a recursive definition of ordered pair which addresses this shortcoming and also naturally generalizes to ordered tuples of greater lenght. There are many advantages to the new definition, for it allows for uniform definitions working (...) equally well in a wide range of models for set theories. In ZFC and closely related theories, the of an ordered pair of two infinite sets under the new definition turns out to be equal to the maximum of the ranks of the sets. (shrink)

A somewhat fictionalized account of several interpretations of implication is presented together with comparisons between classical, modal, tense, and intuitionistic logics.

This thesis is intended t0 help develop the theory 0f coalgebras by, Hrst, taking classic theorems in the theory 0f universal algebras amd dualizing them and, second, developing an interna] 10gic for categories 0f coalgebras. We begin with an introduction t0 the categorical approach t0 algebras and the dual 110tion 0f coalgebras. Following this, we discuss (c0)a,lg€bra.s for 2. (c0)monad and develop 2. theory 0f regular subcoalgebras which will be used in the interna] logic. We also prove that categories 0f (...) coalgebras are completc, under reasonably weak conditions, and simultaneously prove the wellknown dual result for categories 0f algebras. We dose the second chapter with 2. discussion 0f bisimulations in which we introduce a weaker 110tion 0f bisimulaticn than is current in the literature, but which is w€H—b€ha.v€d and reduces t0 the standard defmition under the assumption 0f choice. (shrink)

This work is a step toward the development of a logic for types and computation that includes not only the usual spaces of mathematics and constructions, but also spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes that we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we first study axiomatically, and then by deriving a modal calculus as its internal logic. The resulting (...) framework is intended as a setting for the logical and categorical study of relative computability. (shrink)