Basic Universe is Inductive
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Let $V$ be a basic universe.
Then $V$ is an inductive class.
By definition of basic universe, $V$ is a class containing all sets as elements.
By the Axiom of the Empty Set:
- the empty class $\O$ is a set.
Hence $\O$ is an element of $V$.
By the Axiom of Powers, if $x$ is a set, then $\powerset x$ is a set.
By definition of power set:
- $\set x \subseteq \powerset x$
By the Axiom of Swelledness, $\set x$ is a set.
By the Axiom of Unions, if $x$ is a set, then $\bigcup x$ is a set.
Thus $x \cup \set x$ is a set.
The result follows.
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 2$ Definition of the Natural Numbers