Cowen's Theorem/Lemma 7

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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 $g$ relative to $x$.


Proof

Let $x \in M$.

Then $x \in M_x$.

Because $x \subseteq x \cup \map g x$, we have from Lemma $3$:

$M_x \subseteq M_{x \cup \map g x}$

Hence:

$x \in M_{x \cup \map g x}$

Also:

$\map g x \subseteq x \cup \map g x$

and so:

$\map g x \in M_{x \cup \map g x}$

Also, from Lemma $6$:

$M_{x \cup \map g x} \subseteq M$

Hence:

$\map g x \in M$

and the result follows.

$\blacksquare$


Source of Name

This entry was named for Robert H. Cowen.


Sources