Peano arithmetic

Peano arithmetic is an axiomatic theory over the natural numbers.

Axioms

 * 1) \(0 \in \mathbb{N}\): Zero is a natural number.
 * 2) \(\forall x \in \mathbb{N}: \exists y: y = Sx\): Every natural number has a successor.
 * 3) \(\not \exists x: 0 = Sx\): Zero has no predecessor.
 * 4) \(\forall x, y \in \mathbb{N}: x = y \Leftrightarrow Sx = Sy\): Two numbers are equal iff their successors are equal.
 * 5) \(\forall M: M \subseteq \mathbb{N} \wedge 0 \in M \wedge \forall x \in M: (Sx \in M) \Rightarrow M = \mathbb{N}\): \(\mathbb{N}\) contains zero and the successor to each of its own members. Any subset of \(\mathbb{N}\) with these properties is precisely \(\mathbb{N}\).

We can define a standard suite of operations used in Peano arithmetic:


 * Addition: \(a + 1 = Sa\) and \(a + Sb = S(a + b)\)
 * Multiplication: \(a \times 1 = a\) and \(a \times Sb = (a \times b) + a\)