Lecture Notes for 427: Part 1

We write Z for the set of integers, and N = {1, 2, 3, . . . } for the set of natural numbers. I’ll use M 28 Aug the convention where natural numbers are assumed to be positive. I may also write Z>0 for N. We also use the notation Z≥0 = {0, 1, 2, . . . } for the set of non-negative integers. Well-ordering principle (WOP). Every non-empty subset of N contains a smallest element. Explicitly: if S ⊆ N and S 6= ∅, then there exists a ∈ S such that for all s ∈ S we have a ≤ s. Note: if S = ∅, there is no smallest element of S, since there are no elements in S at all (no way to have s ∈ S). Note: If we replace N with Z, R, or [0, 1], none of these satisfy the well-ordering property. The well-ordering principle is equivalent to mathematical induction. Mathematical induction (MI). Let S ⊆ N be a subset such that (i) 1 ∈ S, and (ii) for all k ∈ N, if k ∈ S then also k + 1 ∈ S. Then S = N. This is often phrased in terms of a sequence of propositions as follows: Let P1, P2, . . . be a sequence of statements such that (i) P1 is true, and (ii) for all k ∈ N, if Pk is true then Pk+1 is true. Then Pn is true for all N. To see that this is really the same as mathematical induction, consider the set S := {n ∈ N | Pn is true }.