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"Axioms should be minimal" is a fine statement which doesn't reflect the way mathematicians actually work. For instance, the commonly accepted list of axioms for a vector space is not minimal; nor is the set of axioms for a group. Nevertheless, nearly all sources define groups with the non-minimal and symmetric set of axioms rather the minimal and obscure one. The same is true about the Peano axioms. It's possible to present a minimal version of them; yet these are not Peano's axioms as they're normally presented in mathematical texts. One very good reason to retain the axiom "every natural number except 0 has a predecessor" (which is in fact the commonly accepted form of this axiom) is that it's common to treat the induction axiom as a special and very strong axiom, and to study fragments of Peano arithmetic defined by other axioms without induction, or with weaker forms of it. -- AV |
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