Formally (following the mainstream in math) the numbers are constructed from scratch out of the axioms of Zermelo Fraenkel set theory (a.k.a. ZF set theory) [Enderton77, Henle86, Hrbacek84]. The only things that can be derived from the axioms are sets with the empty set at the bottom of the hierarchy. This will mean that any number is a set (it is the only thing you can derive from the axioms). It doesn't mean that you always have to use set notation when you use numbers: just introduce the numerals as an abbreviation of the formal counterparts.

The construction starts with *N* and algebraically speaking, *N* with its
operations and order is quite a weak structure. In the following constructions
the structures will be strengthen one step at the time: *Z* will be an
integral domain, *Q* will be a field, for the field *R* the order will be
made complete, and field *C* will be made algebraically complete.

Before we start, first some notational stuff:

- a pair ,
- an equivalence class ,
- the successor of
*a*is .

Mon Feb 23 16:26:48 EST 1998