Class of All Ordinals is Ordinal

This page has been identified as a candidate for refactoring of advanced complexity. In particular: Going to restructure this entire argument: is or is not $\On$ an ordinal? as a philosophical position like AoC and LEM and so on. Not tonight, I need to mull it more. Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

Theorem

The class of all ordinals $\On$ is an ordinal.

This article, or a section of it, needs explaining. In particular: Does this not contradict Class of All Ordinals is Proper Class? Sorry, I now read the discussion page. But the current Definition:Ordinal requires an ordinal to be a set, so $\On$ isn't an ordinal. We need a scond definition of ordinal for this theorem. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Proof

The validity of the material on this page is questionable. In particular: This whole proof probably needs to be rewritten from scratch. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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

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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:

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

This article, or a section of it, needs explaining. In particular: 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. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

The initial segment of the class of all ordinals is:

$\set {x \in \On : x \subset \On}$

This article, or a section of it, needs explaining. In particular: What is meant by $\subset$ here: $\subseteq$ or $\subsetneqq$? Initial segment makes no sense here because $\On$ is not a set. 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. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

This class is equal to $\On$.

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

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $7.12$