Are the ordinals a set or a proper class?

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SUMMARY

The discussion centers on the classification of ordinals in set theory, specifically whether they constitute a set or a proper class. The Burali-Forti paradox establishes that ordinals cannot form a set, as they are an infinite collection that leads to contradictions in naive set theory. The distinction between sets and proper classes is clarified, emphasizing that sets can contain other sets, while proper classes cannot. This understanding is crucial for navigating foundational issues in set theory.

PREREQUISITES
  • Understanding of set theory concepts, particularly "sets" and "proper classes."
  • Familiarity with the Burali-Forti paradox and its implications.
  • Knowledge of naive set theory and its limitations, including the Russell Paradox.
  • Basic comprehension of ordinals and their construction from the null set.
NEXT STEPS
  • Research the Burali-Forti paradox in detail to understand its implications on set theory.
  • Study the Russell Paradox and its impact on naive set theory.
  • Explore the definitions and properties of ordinals in formal set theory.
  • Examine the distinctions between sets and proper classes in advanced mathematical contexts.
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Mathematicians, logicians, and students of set theory looking to deepen their understanding of the foundational concepts of sets and classes.

meteor
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Do the ordinals form a set?
I'm confused, I thought that they form a set, but the Burali-Forti paradox says that they are not a set, but instead a proper class.
I always thought that a set was a finite or infinite collection of things. If the ordinals are an infinite collection of things, I do not see why they can't form a set
 
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Take a close look at the Burali-Forti paradox, especially at their definition of "set". In "naive set theory" a set is any collection of things but that leads to problems (in particular, the Russell Paradox)- that's why it's called "naive". As soon as you start talking about "proper classes" you are using the rule that a "set" cannot have sets as members.
 
Sets certtaily can have sets as members. The ordinals are build up from the null set. Zero is defined as the null set and one is {0}. We have the null set as an element of one.

A set can be on the left or the right side of 'is an element of". A class can only be on the right of 'is an element of'.
 

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