Peano’s Axioms
Numerous formulations of the axioms of arithmetic exist, but the ones that
mathematicians and logicians use were devised by Giuseppe Peano, an Ital-
ian mathematician of the late nineteenth and early twentieth century. His
axioms for the natural numbers (another term for the positive integers) were
Axiom 1: The number 1 is a natural number.
Axiom 2: If a is a natural number, so is a1.
Axiom 3: If a and b are natural numbers with ab, then a 1 b1.
Axiom 4: If a is a natural number, then a 1 1.If these were the only axioms, not only would you be able to balance your
checkbook, but mathematicians would have no difficulty showing that the
axioms were consistent. It was Peano’s fifth axiom that caused the problems.
Axiom 5: If S is any set that contains 1, and has the property that if a
belongs to S, so does a 1, then S contains all the natural numbers.
This last axiom, sometimes called the principle of mathematical induc-
tion, allows mathematicians to prove results about all natural numbers.
Suppose that one day you find yourself at a boring meeting, and with
nothing better to do you start jotting down sums of odd numbers. After a
short while you have compiled the following table:
1 1
1 3 4
1 3 5 9
1 3 5 7 16Suddenly, you notice that all the numbers on the right are squares, and
you also notice that the number on the right is the square of the number
of odd numbers on the left. This leads you to form the following conjec-
ture: the sum of the first n odd numbers (the last of which is 2n1) is n^2.
You can write this as a single formula:
1 3 5 (2n1)n^2.So how are you going to prove this? There are at least two cute ways to do
this. The first is an algebraic version of Gauss’s trick. Write down the
sum S in both increasing and decreasing order.
S 1 3 (2n3) (2n1)
S(2n1) (2n3) 3 1Each sum contains precisely n terms, so if we add the left sides of both
equations we get 2S, and by looking at the sums of each column, we no-
120 How Math Explains the World