A description/proof of that unbounded collection of ordinal numbers is not set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of unbounded collection.
- The reader knows a definition of ordinal number.
- The reader admits the Burali-Forti theorem.
Target Context
- The reader will have a description and a proof of the proposition that any unbounded collection of ordinal numbers is not any set.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
Any unbounded collection, \(O\), of ordinal numbers, ordered by the \(\epsilon\) ordering, is not any set.
2: Proof
\(\cup O\) is the collection of all the ordinal numbers, because for any ordinal number, \(o_1\), there is an ordinal number, \(o_2 \in O, o_1 \in o_2\), so, \(o_1 \in \cup O\). By the Burali-Forti theorem, \(\cup O\) is not any set. So, \(O\) is not any set, because if \(O\) was a set, \(\cup O\) would a set by the union axiom.
