Cowen's Theorem/Lemma 4

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:

$M$ is closed under chain unions.

Proof

Let $C$ be a chain of elements of $M$.

For each $x \in C$, we have $x \subseteq \ds \bigcup C$.

Hence by Lemma $3$:

$M_x \subseteq M_{\mathop \cup C}$

Also, because $x \in M_x$, we have:

$x \in M_x$

Hence:

$x \in M_{\mathop \cup C}$

Thus:

$C \subseteq M_{\mathop \cup C}$

Because $M_{\mathop \cup C}$ is closed under chain unions:

$\ds \bigcup C \in M_{\mathop \cup C}$

Hence:

$\ds \bigcup C \in M$

$\blacksquare$

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 $7.7$