The Scope of First-Order Logic

Abstract

This chapter discusses what has been achieved so far and what implications this has for the foundations of mathematics. Among others, two questions are treated: First, mathematical proofs were carried out before the notion of proof was made precise, a vicious circle? Second, in view of the deficiencies in expressive power of first-order logic (as shown in the previous chapter), what effect does the restriction to first-order logic have on the scope of the investigations? Answering these questions, the development of present-day mathematics in the framework of first-order logic is outlined, leading to the introduction of the Zermelo-Fraenkel axioms for set theory.