ABSTRACT
This is a preparatory chapter, intended for students who have not much experience in dealing with axiomatic theories. The chapter starts with some drops of the early history of set theory as it was founded by the work of Cantor and Dedekind, until the appearance of the antinomies by the beginning of the XXth century. Some overview of axiomatic theories is then given, distinguishing between intuitive theories and formal theories; the differences between these two approaches is exemplified by considering Euclid's Elements versus Hilbert's Foundations of Geometry. The role of primitive notions and axioms in an axiomatic theory is also discussed. Finally, Zermelo's presentation of an axiomatic foundation for set theory that avoids the antinomies while preserving the important results of the theory of his time, is also briefly considered.
The last two sections deal with some preliminaries for the first part of the book. Namely, the meaning and use in Mathematics of the logical connectives is explained in the first of those sections. The second displays the material on permutations that will be needed later; it discusses at the most elementary level the product of permutations, and obtains a generating set for the permutations of three objects.
