This chapter contains the second part of our presentation of set notation. It is devoted to the study of the construction of mathematical objects: that is, finite sets, natural numbers, finite sequences, and finite trees. All such constructions are realized by using the same inductive method based on the fixpoint theorem of Knaster and Tarski. We shall also describe the way recursive functions on these objects can be formally defined.

At the beginning of the chapter generalized intersection and union are presented, then we introduce the framework that we shall use in order to construct our objects. Then the construction of each of the mentioned objects follows in separate sections. Finally, at the end of the chapter, we present the concept of well-foundedness, which unifies completely each of the previous constructs.

Hurried readers, who are not interested in the details of these formal constructions, can skip them. They can go to Appendix A, where the notations are all listed.

Generalized Intersection and Union

Before presenting the framework that will allow us to construct mathematical objects in a systematic manner (section 3.2), we need first to introduce two important set theoretic concepts called generalized intersection and generalized union.

As usual, we first introduce our concepts syntactically, then we give an informal explanation and some examples, and finally we state and prove their basic properties. We now present the syntax of the new constructs.