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Proof: (related to "Equivalent Notions of Ordinals")

For a better readability, we will write in the following $\in$ instead of $\in_X$ but mean the contained relation $\in_X$ defined on $X.$

$(1) \Rightarrow (2)$

  • Let $X$ be an ordinal.
  • By definition, $(X,\in)$ is transitive and strictly and totally and well-ordered by $\in_X.$
  • Let $u \in v \in w\in X.$
  • Because $X$ is transitive, $u\in v\in X.$
  • Because $X$ is transitive, $u\in X.$
  • For the same reason, $u\in w.$
  • Therefore, every element $w\in X$ is itself a transitive set.

$(2) \Rightarrow (1)$

$(2) \Rightarrow (3)$

$”\Rightarrow”$:

  • Let $w\in X.$
  • From $(2)$ it follows that $w\in X$ is itself a transitive set. It remains to be shown that $w\subset X$ (i.d. that $w$ is a proper subset of $X.$)
  • If $v\in W$ then $v\in X,$ therefore $w\subseteq X.$
  • But $w\neq X$ since otherwise $X\in X,$ in contradiction to the axiom of foundation.

$”\Leftarrow”$:

  • Let $w\subset X$ be itself a transitive set.
  • Since $w$ is a proper subset of $X,$ the set difference $X\setminus w$ is not empty.
  • Since $(2)$ and $(1)$ are equivalent, $\in$ is a strict order and a well-order.
  • Therefore, $X\setminus w$ contains a minimal element $z\in X\setminus w.$
  • Thus, $z\in X$ and $z\not\in w$ and there is no element $x\in X\setminus w$ with $x\in z.$ We will show that $z=w,$ which means $w\in X.$
    • We have $z\subseteq w.$
      • If $y\in z,$ then since $y\in z\in X$ we have $y\in X,$ because $X$ is transitive.
      • But $z$ is minimal in $X\setminus w$, i.e. there is no other element $x\in X\setminus w$ with $x\in z.$
      • Since $y\in z$ and $y\in X$ and there is no other element $x\in X\setminus w$ with $x\in z,$ we have $y\in w.$
    • We have $w\subseteq z.$
      • Let $y\in w.$
      • We have $z\not\in y,$ otherwise we would have $z\in w$ because $w$ is transitive, in contradiction to the hypothesis $z\not\in w.$
      • Therefore, $y\neq z$ and $z\not\in y$ which means $y\in z,$ since $\in$ is a strict total order.
  • Altogether, we have shown, given the equivalence $(1)\Leftrightarrow (2),$ that from $w$ being transitive and $w\subset X$ it follows that $w\in X.$

$(3) \Rightarrow (2)$

  • Let $w\in X$ if and only if $w\subset X$ and $w$ is transitive for all elements $w\in X.$
  • Therefore, all elements $w\in X$ are transitive sets.
q.e.d

| | | | created: 2019-03-08 11:15:24 | modified: 2019-03-08 12:24:48 | by: bookofproofs | references: [656], [8055]


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Bibliography (further reading)

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