[Axioms for] arithmetic

Most of the Peano axioms are straightforward statements of elementary facts about arithmetic. The last axiom is a schema (see page 1156) that states the principle of mathematical induction: that if a statement is valid for a=0, and its validity for a=b implies its validity for a=b+1, then it follows that the statement must be valid for all a. Induction was to some extent already used in antiquity—for example in Euclid's proof that there are always larger primes. It began to be used in more generality in the 1600s. In effect it expresses the idea that the integers form a single ordered sequence, and it provides a basis for the notion of recursion.

In the early history of mathematics arithmetic with integers did not seem to need formal axioms, for facts like x+y==y+x appeared to be self-evident. But in 1861 Hermann Grassmann showed that such facts could be deduced from more basic ones about successors and induction. And in 1891 Giuseppe Peano gave essentially the Peano axioms listed here (they were also given slightly less formally by Richard Dedekind in 1888)—which have been used unchanged ever since. (Note that in second-order logic—and effectively set theory— + and × can be defined just in terms of Δ; see page 1160. In addition, as noted by Julia Robinson in 1948 it is possible to remove explicit mention of + even in the ordinary Peano axioms, using the fact that if c==a+b then (Δa × c) × (Δb × c) == Δ(c × c) × (Δa × b). Axioms 3, 4 and 6 can then be replaced by a × b == b × a, a × (b × c) == (a × b) × c and (Δa) × (Δa × b) == Δa × (Δb × (Δa)). See also page 1163.)

The proof of Gödel's Theorem in 1931 (see page 1158) demonstrated the universality of the Peano axioms. It was shown by Raphael Robinson in 1950 that universality is also achieved by the Robinson axioms for reduced arithmetic (usually called Q) in which induction—which cannot be reduced to a finite set of ordinary axioms (see page 1156)—is replaced by a single weaker axiom. Statements like x + y==y + x can no longer be proved in the resulting system (see pages 800 and 1169).

If any single one of the axioms given for reduced arithmetic is removed, universality is lost. It is not clear however exactly what minimal set of axioms is needed, for example, for the existence of solutions to integer equations to be undecidable (see page 787). (It is known, however, that essentially nothing is lost even from full Peano arithmetic if for example one drops axioms of logic such as Not[Not[a]] == a.)

A form of arithmetic in which one allows induction but removes multiplication was considered by Mojzesz Presburger in 1929. It is not universal, although it makes statements of size n potentially take as many as about 2^{2n} steps to prove (though see page 1143).

The Peano axioms for arithmetic seem sufficient to support most of the whole field of number theory. But if as I believe there are fairly simple results that are unprovable from these axioms it may in fact be necessary to extend the Peano axioms to make certain kinds of progress even in practical number theory. (See also page 1166.)