Cardinality of the set of ordinal numbers

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Discussion Overview

The discussion centers around the cardinality of the set of ordinal numbers, including both transfinite and finite ordinals. Participants explore the nature of ordinals in the context of set theory, particularly focusing on whether they can be considered a set and the implications of that classification.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant inquires about the cardinality of ordinal numbers and expresses interest in proofs related to transfinite numbers.
  • Another participant asserts that ordinal numbers do not form a set but rather a "proper class," suggesting that they lack a well-defined cardinality.
  • A different participant acknowledges that ordinals are uncountably infinite and seeks to understand the nature of their uncountability.
  • Further elaboration is provided on the distinction between naive and axiomatic set theory, emphasizing that axiomatic set theory imposes limitations on what can be considered a set, with a reference to the Burali-Forti paradox as an argument against ordinals forming a set.
  • One participant expresses enjoyment of the topic without contributing additional claims.

Areas of Agreement / Disagreement

Participants express differing views on whether ordinal numbers can be classified as a set, with some asserting they form a proper class and others questioning the implications of that classification. The discussion remains unresolved regarding the cardinality of ordinals.

Contextual Notes

The discussion highlights the limitations of definitions in set theory and the implications of axiomatic frameworks, particularly concerning the classification of ordinals and their cardinality.

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 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.
 
AlephOmega said:
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 [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.
 
Oh. Fun stuff.
 

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