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
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Cantor's paradox
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Cantor's paradox
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.