Axiom 2.1. 0 is a natural number. Axiom 2.2. If n is a natural number, then n+ is also a natural num- ber. Definition 2.1.3. We define 1 to be the number 0+, 2 to be the number (0++)+, 3 to be the number ((0++)++)+, etc. (In other words, 1 = 0+, 2 := 1+, 3 := 2+, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.) Axiom 2.3.0 is not the successor of any natural number; i.e., we have n+ 0 for every natural number n. Axiom 2.4. Different natural numbers must have different successors; i.e., if n, m are natural numbers and n‡m, then n‡m+. Equiv- alently², if n+ = m+, then we must have n = m. Example 2.1.9. (Informal) Suppose that our number system N con- sisted of the following collection of integers and half-integers: N = {0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, ...}. (This example is marked "informal" since we are using real numbers, which we're not supposed to use yet.) One can check that Axioms 2.1- 2.4 are still satisfied for this set.