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Corollary: Strictly, Well-ordered Sets and Transitive Sets

For every strictly-ordered set $(V,\prec),$ which is a well-order, there is a unique transitive set $(X,\in_X)$ with the following properties:

  • $(X,\in_X)$ is also a strictly-ordered and well-ordered set with respect to the contained relation $\in_X$
  • Between the two orders $(V,\prec)$ and $(X,\in_X)$ there is an order embedding $\pi:V\to X,$ i.e. an injective function $\pi$ fulfilling the property $u\prec v\Longleftrightarrow \pi(u)\in_X\pi(v).$

| | | | | created: 2019-03-07 15:23:59 | modified: 2019-03-07 16:29:47 | by: bookofproofs | references: [8055]

1.Proof: (related to "Strictly, Well-ordered Sets and Transitive Sets")


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Bibliography (further reading)

[8055] Hoffmann, D.: “Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise”, Hoffmann, D., 2018

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