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

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 13$: Arithmetic
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 8$ Definition by finite recursion: Exercise $8.4 \ \text {(a)}$