Definition:Addition in Minimally Inductive Set

Definition
Let $\omega$ be the minimally inductive set.

The binary operation $+$ is defined on $\omega$ as follows:


 * $\forall m, n \in \omega: \begin {cases} m + 0 & = m \\ m + n^+ & = \paren {m + n}^+ \end {cases}$

where $m^+$ is the successor set of $m$.

This operation is called addition.

Also presented as
Some sources, in order to stress the set-theoretical nature of addition being a mapping from $\omega \times \omega$ to $\omega$, express it as:


 * $\forall \tuple {x, y} \in \omega \times \omega: \map A {x, y} = \begin {cases} x & : y = 0 \\ \paren {\map A {x, r} }^+ & : y = r^+ \end {cases}$

from which the given operation emerges when $\map A {x, y}$ is identified with $x + y$.

Also see

 * Addition in Minimally Inductive Set is Unique