Ordinal is Subset of Class of All Ordinals
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Theorem
Suppose $A$ is an ordinal.
Then:
- $A \subseteq \On$
where $\On$ represents the class of all ordinals.
Proof
By Ordinal is Member of Class of All Ordinals:
- $A \in \On \lor A = \On$
In either case:
- $A \subseteq \On$
since $\On$ is transitive.
This article, or a section of it, needs explaining. In particular: Why is $\On$ necessarily transitive? This follows smoothly if it is assumed that $\On$ is a subclass of a Basic Universe, or otherwise from an axiomatic framework. Hence we need to define our axioms here. 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. |
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $7.15$