Formulation Peano axioms

the next 4 axioms describe equality relation. since logically valid in first-order logic equality, not considered part of peano axioms in modern treatments.

the remaining axioms define arithmetical properties of natural numbers. naturals assumed closed under single-valued successor function s.

peano s original formulation of axioms used 1 instead of 0 first natural number. choice arbitrary, axiom 1 not endow constant 0 additional properties. however, because 0 additive identity in arithmetic, modern formulations of peano axioms start 0. axioms 1, 6, 7, 8 define unary representation of intuitive notion of natural numbers: number 1 can defined s(0), 2 s(s(0)), etc. however, considering notion of natural numbers being defined these axioms, axioms 1, 6, 7, 8 not imply successor function generates natural numbers different 0. put differently, not guarantee every natural number other 0 must succeed other natural number.

the intuitive notion each natural number can obtained applying successor sufficiently 0 requires additional axiom, called axiom of induction.

the induction axiom stated in following form:

in peano s original formulation, induction axiom second-order axiom. common replace second-order principle weaker first-order induction scheme. there important differences between second-order , first-order formulations, discussed in section § models below.