# 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