7 - INTUITIONISTIC LOGIC AND TOPOS THEORY

Summary

In which an informal study of formal first-order intuitionistic logic is undertaken. In the first two sections, axioms for the propositional calculus and the predicate calculus are presented, and the use of Kripke models to demonstrate unprovability is illustrated. The last two sections contain two examples of topos models: one sheaf model and one presheaf model, the latter showing the unprovability of the world's simplest axiom of choice.

Intuitionistic prepositional calculus

As we remarked in Chapter 1, the codification of intuitionistic logic, the logic of constructive mathematics, grows out of our mathematical experience. We do not develop the logic in the first instance, and then use that logic as a foundation for our mathematics; rather, we formulate our rules of logic to reflect our mathematical practice. Nevertheless, the abstraction of logical axioms clarifies much of the mathematical experience on which it is based.

Propositional calculus is a formal system for studying the connectives ∨ (or), ∧ (and), ⇒ (implies), and ⇁ (not), which can be used to form new sentences, or propositions, from old ones. The basic ingredients of the propositional calculus are these four connectives and an infinite sequence p, q, r, … of propositional variables.