A description/proof of that descending sequence of ordinal numbers is finite
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of ordinal number.
- The reader admits the proposition that any collection of ordinal numbers has the smallest element.
Target Context
- The reader will have a description and a proof of the proposition that any descending sequence of ordinal numbers is finite.
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 descending sequence, \(o_0, o_1, . . .\) where \(. . . \in o_2 \in o_1 \in o_0\), of ordinal numbers finishes at an \(o_n\).
2: Proof
By the proposition that any collection of ordinal numbers has the smallest element, there is the smallest element of \(\{o_i\}\), which is \(o_n\). Then, there is no \(o_{n + 1}\), because it would be a smaller element, a contradiction.
