Principle E

Theorem

Let $S$ be a set of sets.

Let $S$ be closed under chain unions.

Then every element $b$ of $S$ is a subset of some element of $S$ that has no immediate extension in $S$.

Principle E and Axiom of Choice

Axiom of Choice implies Principle E

Axiom of Choice implies Principle E

Principle E implies Axiom of Choice

Principle E implies Axiom of Choice

Linguistic Note

The term Principle E appears to have been coined by Raymond M. Smullyan and Melvin Fitting for their Set Theory and the Continuum Problem, revised ed.

Its use on $\mathsf{Pr} \infty \mathsf{fWiki}$ is therefore expected be limited to those pages arising directly from concepts raised as a result of that work.

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