Definition:Basic Universe

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A basic universe $V$ is a universal class which satisfies the following axioms:

$\text A 1$: Axiom of Transitivity

$V$ is a transitive class.

$\text A 2$: Axiom of Swelledness

$V$ is a swelled class.

$\text A 3$: Axiom of the Empty Set

The empty class $\O$ is a set, that is:

$\O \in V$

$\text A 4$: Axiom of Pairing

Let $a$ and $b$ be sets.

Then the class $\set {a, b}$ is likewise a set.

$\text A 5$: Axiom of Unions

Let $x$ be a set (of sets).

Then its union $\ds \bigcup x$ is also a set.

$\text A 6$: Axiom of Powers

Let $x$ be a set.

Then its power set $\powerset x$ is also a set.

Also see

  • Results about the basic universe can be found here.