Definition:Minimally Closed Class

From ProofWiki
Jump to navigation Jump to search

Definition

Let $A$ be a class.

Let $g: A \to A$ be a mapping.


Definition 1

$A$ is minimally closed under $g$ with respect to $b$ if and only if:

\((1)\)   $:$   $A$ is closed under $g$             
\((2)\)   $:$   There exists $b \in A$ such that no proper subclass of $A$ containing $b$ is closed under $g$.             


Definition 2

$A$ is minimally closed under $g$ with respect to $b$ if and only if:

\((1)\)   $:$   $A$ is closed under $g$             
\((2)\)   $:$   There exists $b \in A$ such that every subclass of $A$ containing $b$ which is closed under $g$ contains all the elements of $A$.             


Also see

  • Results about minimally closed classes can be found here.


Linguistic Note

The term minimally closed class was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ in accordance with the concept of a minimally inductive class.

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


Both minimally closed class and minimally inductive class are discussed in Set Theory and the Continuum Problem, revised ed. by Raymond M. Smullyan and Melvin Fitting of $2010$.

While the minimally closed class is suggested as a generalisation of the minimally inductive class in an optional section backed up with an exercise, it is not actually named there as such.

The name was coined in order to provide a convenient tag, as a description is unwieldy.