Cantor's Paradox

Paradox

Let $\CC$ be the set of all sets.

Let $\powerset \CC$ denote the power set of $C$.

Is the cardinality of $\CC$ greater than or equal to the cardinality of $\powerset \CC$?

The sets of $\powerset \CC$ must be elements of $\CC$.

Hence:

$\powerset \CC \subseteq \CC$

Hence:

$\card {\powerset \CC} \le \card \CC$

But by Cantor's Theorem:

$\card {\powerset \CC} > \card \CC$

Resolution

This is an antinomy.

The set of all sets is not a set.

This theorem requires a proof. In particular: See Burali-Forti Paradox for similar stuff You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Source of Name

This entry was named for Georg Cantor.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cantor's paradox
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cantor's paradox