Definition:Special Set

Definition

Let $g$ be a progressing mapping.

Let $S$ and $x$ be sets.

We say that:

$S$ is special for $x$ (with respect to $g$)

if and only if:

\((1)\)	$:$							$\O \in S$
\((2)\)	$:$							$S$ is closed under $g$ relative to $x$
\((3)\)	$:$							$S$ is closed under chain unions

Also known as

Instead of $S$ is special for $x$, we can say $S$ is $x$-special.

Also see

• Results about special sets can be found here.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {III}$ -- The existence of minimally superinductive classes: $\S 7$ Cowen's theorem: Definition $7.2$