Sequence of Recursively Defined Terms forms Minimally Inductive Set

Theorem

Let $g$ be a mapping.

Let $\sequence {a_n}$ be a sequence such that:

$a_0 = \O$

and:

$\forall n \in \N: a_{n + 1} = \map g {a_n}$

Then the set:

$\set {a_0, a_1, \ldots, a_n, a_{n + 1}, \ldots}$

is minimally inductive under $g$.

Proof

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Sources

• 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 $8.6$