Ordinal Class is Ordinal
Theorem
The Ordinal Class $\On$ is an ordinal.
Proof
This whole proof probably needs to be rewritten from scratch.
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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:
- $\forall x \in \On: x \subset \On$
We have a problem here since $\On$ is a proper class. The notation $\in $ an $\subseteq$ can naturally be extended to classes, but one must be careful with their use.
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The initial segment of the class of ordinals is:
- $\set {x \in \On : x \subset \On}$
What is meant by $\subset$ here: $\subseteq$ or $\subsetneqq$?
Sorry, but "makes no sense" is a bit strong: "initial segment" has been defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ in the context of a well-ordered set, but the term makes perfect sense if defined in the context of a class.
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Therefore, by the definition of ordinal, $\On$ is an ordinal.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $7.12$