Swelled Set which is Closed under Chain Unions with Choice Function is Type M
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
- $S$ 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 $M$.
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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
- 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 $\text {II}$ -- Maximal principles: $\S 6$ Another approach to maximal principles: Corollary $6.3$