2.1 The Peano Axioms

Axiom 2.1. 0 is a natural number.

Axiom 2.2. If n is a natural number, then n++ is also a natural number.

i.e., 0++ is also a natural number, (0++)++ is also a natural number.

Definition 2.1.3 We define 1 to be the number 0++, 2 to be the number (0++)++， 3 to be the number ((0++)++)++, etc.(In other words, 1:=0++, 2:=1++, 3:=2++, etc. In this text I use "x:=y" to denote the statement that x is defined to equal y.)

Proposition 2.1.4 3 is a natural number.

\(Proof.\) By Axiom 2.1, 0 is a natural number. By Axiom 2.2, 0++=1 is a natural number. By Axiom 2.2 again, 1++=2 is a natural nuber. By Axiom 2.2 again, 2++=3 is a natural number.

Axiom 2.3. 0 is not the successor of any natural number; i.e., we have \(n++ \neq 0\)

Proposition 2.1.6 4 is not equal to 0.

\(Proof.\) By definition, 4=3++. By Axioms 2.1 and 2.2, 3 is a natural number. Thus by Axiom 2.3, \(3++ \neq 0\), i.e., \(4\neq0\).

Axiom 2.4. Different natural numbers must have different successors; i.e., if n,m are natural numbers and n \neq m, then n++ \neq m++. Equivalently, if n++ = m++, then we must have n=m.

This is an example of reformulating an implication using its contrapositive; see Section A.2 for more details. In the converse diretion, if n=m, then n++=m++; this is the axiom of substitution (see Section A.7) applied to the operation ++.

Proposition 2.1.8 6 is not equal to 2.

\(Proof.\) Suppose for sake of contradiction that 6=2. Then 5++=1++, so by Axiom 2.4 we have 5=1, so that 4++=0++. By Axiom 2.4 again we then have 4=0, which contradicts our previous proposition.

Axiom 2.5. (Principle of mathematical induction). Let \(P(n)\) be any property pertaining to a naural number n. Suppose that \(P(0)\) is ture and supose that whenwver \(P(n)\) is true, \(P(n++)\) is also ture. Then \(P(n)\) is ture for every natural nummber n.

Assumption 2.6. (Informal) There exists a number system \(N\), whose elements we will call natural numbers, for which Axioms 2.1-2.5 are ture.

We will make this assumption a bit more precise once we have laid down our notation for sets and functions in the next chapter.

One consquence of the axioms is that we can now define sequences \(recursivly\). Suppose we want to build a sequence \(a_0,a_1,a_2,...\) of numbers by first defining \(a_0\) to be some base value, e.g., \(a_0:=c\) for some number c, and then bu lettimg \(a_1\) be some fnunction of \(a_0\), \(a_1:=f_0(a_0)\), \(a_2\) be some fnunction of \(a_1\), \(a_2:=f_1(a_1)\), and so forth. In geneal, we set \(a_{n++}:=f_n(a_n)\) for some function \(f_n\) from \(N\) to \(N\). By using all the axionms together we will now conclude that this procedure will give a single value to the sequence element \(a_n\) for each ntural number n.

(Strictly speaking, this proposition requires one to define the notion of a \(function\), which we shall do in the next chapter. However, this will not be circular, as the concept of a function does not require the Peano axioms. Proposition 2.1.6 can be formalized more rigorously in the language of set theory; se Exercise 3.5.12.)

Proposition 2.1.6 (Recursive definitions). Suppose for each natural number n, we have some function \(f_n:N \rightarrow N\) from the natural numbers to the natural numbers. Let c be a natural number. Then we can assign a unique natural number \(a_n\) to each number n, such that \(a_0=c\) and \(a_{n++}=f_n(a_n)\) for each natural number n.

\(Proof.\)(Informal) We use induction. we first observe that this procedure gives a single value to \(a_0\), namelt c. (None of the other definitions \(a_{n++}=f_n(a_n)\) will redefine the value of \(a_0\), because of Axiom 2.3.)
Now suppose inductively that the procedure gives a single value to \(a_n\). Then it gives a single value to \(a_{n++}\), namely \(a_{n++}=f_n(a_n)\). (None of the other definitions \(a_{m++}=f_m(a_m)\) will redefine the value of \(a_{n++}\), because of Axiom 2.4.). This completes the induction, and so \(a_n\) is defined for each natural number n, with a single value assigned to each \(a_n\).

Recursive definitions are very powerful; for instance, we can use them to define addition and multiplication, to which we now urn.

2.2 Addition