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Do the ordinals form a set?

  1. Mar 26, 2004 #1
    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
  2. jcsd
  3. Mar 26, 2004 #2


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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.
  4. Mar 26, 2004 #3
    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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