Cowen's Theorem/Lemma 2

Lemma for Cowen's Theorem

Let $g$ be a progressing mapping.

Let $x$ be a set.

Let $\powerset x$ denote the power set of $x$.

Let $M_x$ denote the intersection of the $x$-special subsets of $\powerset x$ with respect to $g$.

Let $M$ be the class of all $x$ such that $x \in M_x$.

We have that:

$\O \in M$

Proof

By Lemma $1$:

$\powerset \O$ is $\O$-special with respect to $g$.

By Power Set of Empty Set:

$\powerset \O = \set \O$

Hence:

$\set \O$ is $\O$-special with respect to $g$.

The only other subset of $\set \O$ is $\O$, which is by definition empty.

So $\set \O$ is the only $\O$-special subset of $\powerset \O$ with respect to $g$.

Hence:

$M_\O = \set \O$

Because $\O \in \set \O$, we have:

$\O \in M_\O$

and so:

$\O \in M$

$\blacksquare$

Source of Name

This entry was named for Robert H. Cowen.

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: Proposition $7.5$