Basic Universe is not Set
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Theorem
Let $V$ be a basic universe.
Then $V$ is not a set.
Proof
Aiming for a contradiction, suppose $V$ were a set.
Then by the the Axiom of Swelledness, $V$ is swelled.
That is, as $V$ is a set, every subclass of $V$ would also be a set.
From Class has Subclass which is not Element, $V$ has a subclass $S$ which is not an element of $V$.
That is:
- $\exists S \subseteq V: S \notin V$
But by definition of a basic universe, $V$ is a universal class.
That is:
- $S \in V$
This contradicts the deduction that $S \notin V$.
Hence the result by Proof by Contradiction.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 2$ Transitivity and supercompleteness