Matthew Burke and Katherine Walton

Foundations of the Real

Number System

Defn: A model of Peano’s Axioms is a set N together with a function f and an object e (N,f,e) such that,

(P1) e Î N.

(P2) The domain of f is in N and for each x Î N, f(x) Î N.

(P3) If x Î N, then f(x) ¹ e.

(P4) If x, y Î N and f(x) = f(y), then x = y.

(P5) If A is a subset of N which contains e and contains f(x) for every x in A then A = N.

Any two models of Peano’s axioms are isomorphic.

Using the standard axioms of ZF set theory, we can construct a model which satisfies the Peano axioms.

Without going into the detains, we will sketch the argument, and give relevant definitions.

We will consider only sets as elements of sets.

Defn: An ordered pair (a,b) is defined as the set {{a}, {a,b}}.

Note (a,a) = {{a}}.

It is true in ZF that if (a,b) = (x,y) then a = x, and b = y.

Defn: Cartesian Product: Given two sets A and B

A x B = {y: y = (a,b) for some a Î A, and for some b Î B}.

Defn: A set R is a binary relation iff every element of R is an ordered pair.

Note that every binary relation is a subset of a Cartesian product A x B where

A = {a: for some b (a,b) Î R}

B = {b: for some a (a,b) Î R}.

A is called the domain of R and B is the image of R.

Defn: An equivalence relation is a relation that is reflexive, symmetric, and transitive.

Defn: given a set x, denote by x⁺ the set x È{x}. x⁺ is called the successor of x.

In ZF no set contains itself, so x⁺ is always distinct from x.

In ZF the null set, Æ, exists by axiom.

We can now define the first few natural numbers as follows:

0 º Æ

1 º Æ⁺ = Æ È{Æ} = {Æ}

2 º Æ⁺⁺ = Æ⁺ È{Æ⁺} = {Æ,{Æ}}.

Before we define all of the natural numbers, we need a few more definitions.

Defn: A set x is a successor set if Æ Î x, and if for each y Î x, y⁺ Î x.

The axiom of infinity asserts the existence of just such a set.

Theorem: There is a minimal successor set, i.e. a successor set that is a subset of every other successor set.

Denote this minimal successor set by w.

Define the natural numbers to be the elements of w.

It is necessary to verify that the Peano Axioms are satisfied by (w, ⁺,Æ).

Define addition for the natural numbers as follows:

" m,n Î w,

m + 0 º m, and

m + n⁺ º (m + n)⁺

Multiplication (note we will often write mn instead of m x n):

" m,n Î w,

m x 0 º 0, and

m x n⁺ º m + (m x n).

Things to show for ‘+’ (note almost all proofs about N are done by induction):

(i) " n Î w 0 + n = n

(ii) " m,n Î w m⁺ + n = (m + n)⁺

(iii) " m,n Î w m + n = n + m

(iv) " m,n,p Î w m + (n + p) = (m + n) + p

Things to show for ‘x’:

(i) " n Î w 0 x n = 0

(ii) " m,n Î w m⁺ x n = (m x n) + n

(iii) " m,n Î w m x n = n x m

(iv) " m,n,p Î w m x (n + p) = (m x n) + (m x p)

(v) " m,n,p Î w m x (n x p) = (m x n) x p

Note special properties of 0 and 1:

0 + m = m = m + 0 " m Î w.

0 x m = 0 = m x 0 " m Î w.

Recall 0⁺ = 1 so m x 1 = m x 0⁺ = m + (m x 0) = m " m Î w.

Similarly, 1 x m = m " m Î w.

So 0 behaves like an additive identity, and 1 behaves like a multiplicative identity.

Note we will now begin using N instead of w to denote the natural numbers.

Defn: " m, n Î N, m < n iff $ x Î N with x ¹ 0 s.t.

m + x = n.

Defn: " m, n Î N, m £ n iff $ x Î N s.t.

m + x = n.

Properties of N:

(i) " n Î N either n = 0, of $ m Î N s.t. n = m⁺.

(ii) " m, n Î N if m ¹ 0, then m + n ¹ 0.

(iii) " m, n Î N if m ¹ 0 and n ¹ 0, then m x n ¹ 0.

(iv) " n Î N, n⁺ = n + 1.

(v) " m, n Î N, if m < n then m⁺ £ n.

(vi) " m, n, p Î N, if m + n = p + n, then m = p (cancellation of addition).

Defn: A set A of natural numbers has a least member iff $ n Î A s.t. n £ m " m, n Î A. If A has a least member, then it is n.

Well Ordering Principle: Every non-empty set of Natural numbers has a least member.

Integers

Let a, b, c, d Î N. Then define an equivalence relation on N by

(a,b) ~ (c,d) iff a + d = b + c

Intuitively we have a – b = c – d iff a + d = b + c.

Then we define the integers to be the equivalence classes under ~.

Ex. The set {(a,b): a + 1 = b} is the integer –1. Note (2,3) Î –1, and 2 – 3 = –1.

We will use (a,b) to denote the equivalence class determined by (a,b).

Define ‘Å’ in the integers by

(a,b) Å (c,d) º (a + b,c +d)

Define ‘Ä’ in the integers by

(a,b) Ä (c,d) º (ac + bd,ad +bc)

Verify Å and Ä are well defined.

To Show: Commutative, and Associative properties of Å.

Commutative, Associative, and Distributive properties of Ä.

Note the special properties of (0,0), and (1,0).

(0,0) is the additive identity in the integers.

(1,0) is the multiplicative identity in the integers.

Denote the integers by Z.

N is embedded in Z. (m,0) Î Z behaves like m Î N.

Additive Inverses:

Given an arbitrary integer (a,b) note that (a,b) Å (b,a) = (a + b,b + a) = (0,0)

We write – (a,b) for (b,a), and we abbreviate (a,b) Å (-(c,d)) by (a,b) - (c,d).

To Show:

(i) (–(a,b)) Ä (c,d) = -((a,b) Ä (c,d)).

(ii) (–(a,b)) Ä (-(c,d)) = (a,b) Ä (c,d).

Defn: An integer (a,b) is positive iff b < a in N.

We need to verify that positive integers are well defined.

Denote the set of positive integers by Z⁺.

More properties of Z:

(i) " x Î Z, either x Î Z⁺, x = 0, or x Î Z⁺.

(ii) " x,y Î Z⁺ x Å y Î Z⁺.

(iii) " x,y Î Z⁺ x Ä y Î Z⁺.

(iv) " x,y Î Z, if x ¹ 0 and y ¹ 0, then x Ä y ¹ 0.

Thus Z is an integral domain.