The previous chapter was somewhat informal. Starting in this chapter we develop the subject systematically and (usually) in logical order. This does not mean that every step in every chain of reasoning will be written out and referred back to the axioms or to results that have already been established. Such a procedure, though possible, is extremely tedious. The goal, rather, is to include enough results – and enough examples of reasoning – so that it may be clear how the gaps might be filled.

The Real Numbers

Our starting point is the real number system ℝ. This is a set that has two algebraic operations, addition and multiplication, and an order relation <. Let a, b denote arbitrary elements of ℝ. Addition associates to any pair a, b a real number denoted a + b, while multiplication associates a real number denoted a · b or simply ab. That < is a relation simply means that certain ordered pairs (a, b) of elements of ℝ are selected, and for these pairs (only) we write a < b. These operations and the order relation satisfy the following axioms, or conditions, in which a, b, c denote arbitrary elements of ℝ.

1. A1 (a + b) + c = a + (b + c).

2. A2a + b = b + a.

3. A3There is an element 0 such that, for all a, a + 0 = a.

4. A4For each a ∈ ℝ there is an element –a ∈ ℝ such that a + (–a) = 0.

5. M1 (ab)c = a(bc).

6. M2ab = ba.

7. M3There is an element 1 ≠ 0 in ℝ such that, for all a, a · 1 = a.

8. […]