Swelled Set which is Closed under Chain Unions with Choice Function is Type M

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Theorem

Let $S$ be a set of sets which:

is closed under chain unions
has a choice function $C$ for its union $\ds \bigcup S$.


Then:

$S$ is swelled

if and only if:

$S$ is of type $M$.


Proof

Sufficient Condition

Let $S$ be swelled.

Let $b \in S$ be arbitrary.

From Closure under Chain Unions with Choice Function implies Elements with no Immediate Extension:

$b$ is the subset of an element of $S$ which has no immediate extension in $S$.

Let $x \in S$ have no immediate extension in $S$.

Then from Element of Swelled Set with no Immediate Extension is Maximal, $x$ is a maximal element under the subset relation on $S$.

That is:

$b$ is the subset of a maximal element of $S$ under the subset relation.

As $b$ is arbitrary:

every element of $S$ is a subset of a maximal element of $S$ under the subset relation.

Hence, by definition, $S$ is of type $M$.

$\Box$


Necessary Condition

Let $S$ be of type $M$.


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Historical Note

Smullyan and Fitting, in their Set Theory and the Continuum Problem, revised ed. of $2010$, demonstrate only the sufficient condition: that a swelled set with the given criteria is of type $M$.

In raising the question as to whether such a set of sets which is not swelled would also be of type $M$, they say:

We posed this question to Professor Herman Rubin, who informed us that it would not.


Sources

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