Question

Using field axioms and order axioms prove the following theorems

(i) The sets R (real numbers), P (positive numbers) and [1, infinity) are all inductive

(ii) N (set of natural numbers) is inductive. In particular, 1 is a natural number

(iii) If n is a natural number, then n >= 1

(iv) (The induction principle). If M is a subset of N (set of natural numbers) then M = N

The following definitions are given:

A subset S of R is called inductive, if 1 is an element of S and if x + 1 is an element of S whenever x is an element of S.

The intersection of all inductive sets if called the set of natural numbers and is denoted by N.

Answer 1