Peano axioms
The natural numbers form a set containing together with a successor function such that , implies , and every subset of containing and closed under is all of .
Axiom 3.1 (Peano Axioms)
-
[ is not the successor of anything.]
-
, .
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Let be a subset of such that
- , and
- ,
then .
Arithmetic from the axioms
Addition is defined recursively by and , and multiplication by and . Induction shows that these operations satisfy the usual rules of arithmetic: and are commutative and associative, and distributes over .
Individual facts are proved the same way. The computation
shows directly. For general , one inducts on with base case , where the claim becomes .
Related
Stated in
- Axiom 3.1 (Peano Axioms)§3.1 Construction of