Description: For forms Form 144 and Form 179-\(\epsilon\)
Content:
For forms Form 144 and Form 179: Let \(W(0)\) be the class of well orderable sets and let\(W(\alpha + 1) = \{\,\bigcup Q: Q\subseteq W(\alpha)\) and\(Q\in W(0)\}\) and \(W(\lambda ) = \bigcup\{ W(\beta): \beta <\lambda\}\) for limit ordinals \(\lambda\). Then \(W\) (the class of almost well orderable sets) is defined to be \(\bigcup\{W(\alpha): \alpha\in On\}\). See Keisler [1970]. Blass [1977] has shown that If \(P\subseteq W\) and \(P\in W\), then \(\bigcup P\in W\). Regarding the Form 179-\(\epsilon\) \((\forall x, x\in W(\alpha))\), David [1980] shows that for ordinals \(\alpha \) and \(\beta\) with \(0<\alpha<\beta\), the statement \(\forall x\), \(x \in W(\beta)\) does not imply\(\forall x\), \(x \in W(\alpha )\). (Clearly, \(\forall x,\ x \in W(\alpha )\) implies \(\forall x,\ x \in W(\beta ).)\) It is clear that Form 30 + Form 294 (every linearly ordered \(W\)-set can be well ordered) implies that every \(W\)-set can be well ordered. Thus, Form 30 + Form 294 + Form 144 (every set is a \(W\)-set) implies \(AC\).
Howard-Rubin number: 25
Type: Definitions and equivalencies
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