In 1889, the Italian mathematician Giuseppe Peano (1858-1932) published the first set of axioms, or assumptions, upon which to build a rigorous system of arithmetic, number theory, and algebra. Peano was a professor and developer of advanced mathematics, but he had become troubled that the grand edifice of mathematics built up over the course of 2000 years rested on the shakiest of foundations, namely, ordinary arithmetic. Sure, every school child knew the rules of counting, addition, subtraction, multiplication, and division; but whence came these "rules"? What reason do we have to believe that these rules are valid. Euclid (c. 300 BC) had placed a firm foundation under plane geometry with his set of five axioms, from which the whole of classical geometry could be derived deductively. This was Peano's goal for arithmetic. To reach this goal, he began, as Euclid did, with five axioms:

Axiom 1. 0 is a number.

Axiom...