Cardinality of the set of ordinal numbers

In summary, the question is about the cardinality of the set of ordinal numbers, but it is not possible to have a set of ordinal numbers due to the Burali-Forti paradox. Therefore, the ordinal numbers do not have a cardinality and the concept of making a set out of anything does not apply in axiomatic set theory. A simplified proof for this concept is also requested.
  • #1
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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  • #2
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.
 
  • #3
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.
 
  • #4
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 [itex]\bigcup{A}[/itex] is an ordinal [itex]\alpha[/itex]. But then [itex]\alpha+1[/itex] 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.
 
  • #5
Oh. Fun stuff.
 

1. What is the definition of the cardinality of the set of ordinal numbers?

The cardinality of the set of ordinal numbers is the number of elements in the set, or the size or magnitude of the set. It is a measure of the number of distinct objects in the set.

2. How is the cardinality of the set of ordinal numbers related to the concept of size?

The cardinality of the set of ordinal numbers is a way to quantify the size of the set. It tells us how many distinct elements are in the set, giving us an idea of its magnitude.

3. Can the cardinality of the set of ordinal numbers be infinite?

Yes, the cardinality of the set of ordinal numbers can be infinite. In fact, the cardinality of the set of ordinal numbers is always infinite, as there is no largest ordinal number. It is a continuous sequence that goes on indefinitely.

4. How is the cardinality of the set of ordinal numbers different from the cardinality of other sets?

The cardinality of the set of ordinal numbers is different from the cardinality of other sets because it represents the order of the elements in the set, rather than just the number of elements. This means that the same set of elements can have different cardinalities if they are arranged in different orders.

5. Can the cardinality of the set of ordinal numbers be compared to the cardinality of the set of natural numbers?

Yes, the cardinality of the set of ordinal numbers can be compared to the cardinality of the set of natural numbers. In fact, the cardinality of the set of natural numbers is equal to the smallest infinite ordinal number, known as aleph-null (ℵ₀). This means that the cardinality of the set of ordinal numbers is greater than or equal to the cardinality of the set of natural numbers.

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