Cowen's Theorem
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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 $x \in M_x$.
Then $M$ is minimally superinductive under $g$.
Proof
Lemma $1$
- $\powerset x$ is $x$-special with respect to $g$.
$\Box$
Lemma $2$
- $\O \in M$
$\Box$
Lemma $3$
- $x \subseteq y \implies M_x \subseteq M_y$
$\Box$
Lemma $4$
- $M$ is closed under chain unions.
$\Box$
Lemma $5$
- $x \subseteq y \implies M_y \subseteq M_x \cup \paren {\powerset y \setminus \powerset x}$
$\Box$
Lemma $6$
- $\forall z: M_z \subseteq M$
$\Box$
Lemma $7$
- $M$ is closed under $g$ relative to $x$.
$\Box$
Recall:
- From Lemma $2$:
- $\O \in M$
- From Lemma $7$:
- From Lemma $4$:
- $M$ is closed under chain unions.
Hence by definition:
- $M$ is superinductive under $g$.
Lemma $8$
Let $A$ be a class which is superinductive under $g$.
Then:
- $M_x \subseteq A$
$\Box$
We have a priori that $M$ is superinductive under $g$.
Let $A$ be superinductive under $g$.
Let $x \in M$.
Then:
- $x \in M_x$
and from Lemma $8$:
- $M_x \subseteq A$
Hence:
- $x \in A$
As $x$ is arbitrary, it follows by definition of subclass that:
- $M \subseteq A$.
Hence by definition $M$ is minimally superinductive under $g$.
$\blacksquare$
Also presented as
Let $g$ be a progressing mapping.
There exists a class $M$ whose elements are exactly those $x$ which are elements of all classes that are superinductive under $g$.
Also:
- $x \in M$ if and only if $x$ is an element of every subset of $\powerset x$ that:
- contains $\O$
- is closed under $g$ relative to $x$
- is closed under chain unions.
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