“An Excursion into Set Theory using a Constructivist Approach”

Constructive logic is an alternative to the theory of classical logic that draws a direct relation between an assertion and its existence. Some of the few important differences between a classical proof and a constructionist proof include the avoidance of proof by contradiction and the rejection of the law of the excluded middle. However, given the possibility that constructive logic is an alternative in proof theory, is this approach to proof an effective alternative to understanding important theorems that have already been classically proven? To motivate an answer, this project shall introduce the basic foundations of Set Theory, a platform upon which we intend to apply constructivist logic. We will then reconstruct important definitions from a constructivist point of view. Informal and Formal Notions of Constructive Mathematics The birth of modern constructive logic is often attributed to the doctoral dissertation of L.E.J Brower (1881-1966) titled Over de Grondslagen der Wiskunde which was published in 1907. In his paper, Brouwer introduced the idea of intuitionism, a fact when applied to mathematics that demands a direct relation between the validity of an abstract mathematical structure to the existence of its construction. To introduce, informally, important concepts in constructive mathematics, one only needs to begin with fundamental concepts of classical logic that form the basis of many theorems in classical mathematics. For example classically, it is possible to prove that holds without explicitly showing that P holds or that Q holds. However, an intuitionist approach to the disjunction rule would require that it is not enough to assert without either proving the existence of P or proving the existence of Q. P Q ∨