Cantor's Paradox
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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.
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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