作者Crazynut

擷文，圖放不下
“Peano axioms” can be found today in numerous textbooks in a form similar to our list in Section 9.2. But the original Peano axioms were quite different. First of all, the notion of a natural number (a member of the set &#8469 was taken as a primitive, and there were explicit axioms stating that zero and its successors are natural numbers. Also Peano used a language richer than first-order logic. There was a single induction axiom, involving quantification over sets, and there were no axioms for addition and multiplication (Example 12.2.2(iv) explains why). Peano’s system can thus be more adequately represented in second-order logic (see Chapters 11 and 12). A minor difference is that his numbers began with 1, not 0.