Peano arithmetic (also known as first-order arithmetic) is a first-order axiomatic theory over the natural numbers.
The language of first-order arithmetic consists of the language of predicate logic extended by the following:
- Constant symbol \(0\), called zero
- Relation symbol \(=\), called equality
- Unary function symbol \(S(x)\), called successor
- Two binary function symbols, \(+(a,b),\cdot(a,b)\), called addition and multiplication respectively, and often denoted by \(a+b,a\cdot b\).
- \(\forall n:0\neq S(n)\) - zero isn't a successor of any natural number.
- \(\forall n,m: S(n)=S(m)\Rightarrow n=m\) - two numbers with equal successors are equal themselves, so \(S(x)\) is an injective function.
- \(\forall n: n+0=n\)
- \(\forall n,m: n+S(m)=S(n+m)\) - this and previous axiom state inductive properties of addition.
- \(\forall n: n\cdot 0=0\)
- \(\forall n,m: n\cdot S(m)=n\cdot m+n\) - this and previous axiom state inductive properties of multiplication.
- For every first-order formula \(\varphi(x)\): \((\varphi(0)\land(\forall n:\varphi(n)\Rightarrow\varphi(S(n)))\Rightarrow\forall n:\varphi(n)\) - this is so called axiom schema of induction, which states that if some property \(\varphi\) holds for zero, and if any number \(n\) posseses this property, then so does its successor, then this property holds for every natural number.
We assume consistency, which is not proven but strongly believed. PA can prove many everyday facts about the natural numbers, and considering Friedman's grand conjecture, it may even be more than enough.
PA has proof-theoretic ordinal \(\varepsilon_0\).