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We assume the existence of certain things called numbers, some of which are called counting numbers, and we take for granted certain statements concerning numbers, called axioms. The first few axioms are
Axiom I
If each of x and y is a number, then x + y (read x plus y) is a number called the sum of x and y. The association with x and y of the sum x + y is called addition.
Axiom II
If each of x, y, and z is a number, then x + (y + z) is (x + y) + z.
Axiom III
0 is a number such that if x is a number, then 0 + x is x.
Axiom IV
If x is a number, then -x is a number such that x + (-x) is 0.
Axiom V
If x and y are numbers, then x + y is y + x.
A suitable question may lead to a theorem and one question may lead to another. For example, a study of Axiom III could suggest the question: Is 0 the only number with the property that if x is a number, then 0 + x is x? It may be shown on the basis of Axioms III and V that the answer is in the affirmative so that we have
Theorem A. If 0′ (read 0 prime) is a number such that if x is a number then 0′ + x is x, then 0′ is 0.
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