Cardinality of the set of ordinal numbers

[AlephOmega]

Does anyone happen to know what the cardinality of the set of ordinal number (transfinite and otherwise) is? A simplified proof would also be much appreciated. Recently I have been very interested in transfinite numbers and the logically gorgeous proofs involved :D

 

[micromass]

The ordinal numbers do not form a set. There are too much ordinal numbers in order for it to be a set. Instead, the ordinal numbers form a "proper class".

Thus, since the ordinal numbers do not form a set, they do not have a cardinality. Likewise, the cardinal numbers do not form a set and thus don't have a well-defined cardinality.

 

[AlephOmega]

Can't you make a set out of anything? I already know the ordinals are uncountably infinite, so I just want it one step farther, "how" uncountably infinite.

PS. I mean the set of the numbers themselves, not the sets they describe.

 

[micromass]

Can't you make a set out of anything? I already know the ordinals are uncountably infinite, so I just want it one step farther, "how" uncountably infinite.

PS. I mean the set of the numbers themselves, not the sets they describe.

No, you can't make a set out of anything! That's the biggest difference between "naive set theory" and "axiomatic set theory". In axiomatic set theory, there are strict limitations on what can be a set and what can't be a set. It turns out that the ordinals do not form a set.

I don't know how much you know about ordinals, but here's an easy argument why the ordinals cannot be a set:

Assume that there exists a set A consisting of all the ordinals. Then $\bigcup{A}$ is an ordinal $\alpha$. But then $\alpha+1$ is an ordinal which is not contained in A.

The above proof is known as the Burali-Forti paradox. It was one of the reasons that axiomatic set theory was developed.

 

[AlephOmega]

Oh. Fun stuff.