Successor of Omega

Theorem

 * $\omega + 1 = \set {0, 1, 2, \ldots; \omega}$

where $\omega$ is the minimally inductive set and $\omega + 1$ is the successor of $\omega$.

Note the use of the semicolon; this is the notation for multipart infinite sets.

Comment
It is customary to use $\omega + 1$ rather than $\omega^+$ for transfinite arithmetic.

However, it needs to be borne in mind that this is not conventional natural number addition.

For example, $\omega + 1 \ne 1 + \omega$.