Definition:Transitive-Closed Class

Definition

Let $K$ be a class

Then $K$ is transitive-closed if and only if:

every transitive subset of $K$ is an element of $K$.

Also known as

Referred to in Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) as a $T$-closed class.

Also see

• Results about transitive-closed classes can be found here.

Linguistic Note

The term Transitive-Closed Class was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.

The concept appears to have been coined by Raymond M. Smullyan and Melvin Fitting under the name $T$-closed class in their Set Theory and the Continuum Problem, revised ed. of $2010$ for the purpose of an exercise.

It is referred to on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a transitive-closed class in order to improve clarity and transparency.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 2$ Ordinals and transitivity: Exercise $2.1$