Zermelo-Fraenkel Set Theory

Chapter 3
Natural Numbers
3.1 Peano Axioms
In order to see that ZF is strong enough to develop all (or most) of mathematics, the
notions and objects from mathematics have to be defined —better: simulated— in ZF.
The main mathematical notion is that of a function, and this one could be introduced
thanks to the possibility of a set-theoretic definition of ordered pair (Definition 2.3.1 p. 8).
As regards the mathematical objects: here follows the (set of) natural numbers; from
these, the other number systems (rationals, reals) may be defined in the well-known way.
The set of natural numbers can be characterized (“up to isomorphism” — see Theorem
3.1) using the Peano Axioms. More precisely, these axioms characterize the system
(IN, 0, S), where S is the successor-operation on IN defined by: S(n) = def n + 1. The Peano
Axioms are the following five statements about this system:
1. 0 is a natural number: 0 ∈ IN,
2. the successor of a natural number is a natural number: n ∈ IN ⇒ S(n) ∈ IN,
3. S is injective: S(n) = S(m) ⇒ n = m,
4. 0 is not a successor: for all n∈IN, it holds that 0 ≠ S(n),
5. (“mathematical”) induction:
if X ⊂ IN is such that (i) 0 ∈ X and (ii) ∀n∈X(S(n) ∈ X), then IN ⊂ X.
Let us call a Peano system any system (A, a 0 , s) that satisfies the Peano axioms.
An isomorphism between such systems (A, a 0 , s) and (B, b 0 , t) is a bijection h : A → B
for which h(a 0 ) = b 0 and such that for all a ∈ A, h(s(a)) = t(h(a)), and systems between
which such an isomorphism exists are called isomorphic. The idea is that isomorphic
systems are complete lookalikes.
Theorem 3.1 Every two Peano systems are isomorphic.
Proof. Sketch. (But see Exercise 31 p. 17.)
Assume that (A, a 0 , s) and (B, b 0 , t) are Peano systems. Define a n = def s(· · · s(a 0 ) · · ·).
(n occurrences of ‘s’).
Claim. a n is different from a 0 , . . . , a n−1 .
Proof. Induction w.r.t. n.
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