Definition:Greatest Set by Set Inclusion/Class Theory

Definition

Let $A$ be a class.

Then a set $M$ is the greatest element of $A$ (with respect to the subset relation) if and only if:

$(1): \quad M \in A$

$(2): \quad \forall S: \paren {S \in A \implies S \subseteq M}$

Also known as

Some sources use the term largest set or largest element. The terms are synonymous.

Also see

• Definition:Smallest Set by Set Inclusion (Class Theory)

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications: Definition $4.8$