Universal Class is Proper
Theorem
Let $V$ denote the universal class.
Then $V$ is a proper class.
Proof 1
Aiming for a contradiction, suppose $V$ is small.
We have that:
- $\operatorname {Ru} \subseteq V$
where $\operatorname {Ru}$ denotes the Russell class.
By Axiom of Subsets Equivalents, $\operatorname {Ru}$ is also small.
This contradicts Russell's Paradox.
$\blacksquare$
Proof 2
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Let $I_V$ be the identity mapping on $V$.
By Identity Mapping is Bijection it follows that $I_V$ is a bijection.
Therefore, by the Axiom of Limitation of Size, $V$ is proper.
$\blacksquare$
Proof 3
Aiming for a contradiction, suppose $V$ is small.
By Cantor's Theorem, there is no surjection from $V$ to $\powerset V$.
By definition of universal class:
- $\powerset V \subseteq V$
By Injection from Subset to Superset, there exists an injection from $\powerset V$ to $V$.
By Injection has Surjective Left Inverse Mapping, there is a surjection from $V$ to $\powerset V$.
But this contradicts Cantor's Theorem.
Therefore $V$ is proper.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 5.23$
- 2002: Thomas Jech: Set Theory (3rd ed.) ... (previous) ... (next): Chapter $1$: Separation Schema