**Proposition**: Equivalent Notions of Ordinals

The following definitions are equivalent:

- $X$ is an ordinal.
- $X$ is a transitive set and all elements $w\in X$ are transitive sets.
- $w\in X$ if and only if $w\subset X$ ($w$ is a proper subset of $X$) and $w$ is transitive.

| | | | | created: 2019-03-08 10:25:52 | modified: 2019-03-08 12:14:24 | by: *bookofproofs* | references: [656], [8055]

## 1.**Proof**: *(related to "Equivalent Notions of Ordinals")*

(none)

[8055] **Hoffmann, D.**: “Forcing, Eine EinfÃ¼hrung in die Mathematik der UnabhÃ¤ngigkeitsbeweise”, Hoffmann, D., 2018

[656] **Hoffmann, Dirk W.**: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011

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