# Does there exist a set of all cardinals? [duplicate]

Does there exist set that contains all the cardinal numbers?

• This has been asked several times before. Once recently. Feb 16, 2013 at 16:39

Assume $C$ was the set of all cardinals. Then $\bigcup C$ would be a cardinal exceeding all cardinals in $C$ which is a contradiction.
• This would also imply that there is no function that maps n to $\aleph_n$? Feb 16, 2013 at 16:43
• @PyRulez The cardinals are not limited to the $\aleph_n$'s, they go way way beyond that. Feb 16, 2013 at 16:43
• Wait, how is that from n to $\aleph_n$? $f(3) \neq \aleph_3$! Feb 16, 2013 at 16:46
Set of cardinals is well ordered by $\in$. Now, as a corollary we get Burali-Forti theorem, which says that there is no the set of all ordinal numbers. As a corollary from a corollary we can prove that, there is no set that contains all the ordinals. Proof: Let $A$ be a set that contains all the ordinals. You can prove that $\{x\in A : x \ \text{is ordinal}\}$ is a set, which contradicts Burali-Forti theorem.