Ordinal Class is Ordinal

From ProofWiki
Jump to navigation Jump to search

Theorem

The Ordinal Class $\On$ is an ordinal.


Proof

NotZFC.jpg

This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.

If you see any proofs that link to this page, please insert this template at the top.

If you believe that the contents of this page can be reworked to allow ZFC, then you can discuss it at the talk page.


The epsilon relation is equivalent to the strict subset relation when restricted to ordinals by Transitive Set is Proper Subset of Ordinal iff Element of Ordinal.

It follows that:

$\forall x \in \On: x \subset \On$



The initial segment of the class of ordinals is:

$\set {x \in \On : x \subset \On}$



This class is equal to $\On$.

Therefore, by the definition of ordinal, $\On$ is an ordinal.

$\blacksquare$


Sources