Cardinality of the set of ordinal numbers

AI Thread Summary
The set of ordinal numbers, including transfinite ordinals, does not form a set but rather a "proper class," which means it lacks a well-defined cardinality. This conclusion arises from axiomatic set theory, which imposes strict limitations on what can be classified as a set. The Burali-Forti paradox illustrates this by showing that if all ordinals were contained in a set, it would lead to contradictions. Consequently, while ordinal numbers are uncountably infinite, they cannot be quantified in terms of cardinality. The discussion emphasizes the distinction between naive and axiomatic set theory in understanding ordinals.
AlephOmega
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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
 
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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.
 
Can't you make a set out of anything? I already know the ordinals are uncountably infinate, so I just want it one step farther, "how" uncountably infinate.

PS. I mean the set of the numbers themselves, not the sets they describe.
 
AlephOmega said:
Can't you make a set out of anything? I already know the ordinals are uncountably infinate, so I just want it one step farther, "how" uncountably infinate.

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.
 
Oh. Fun stuff.
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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