Description: Notes on \(UT(WO,\kappa ,WO),\ C(WO,\kappa )\) and \(D_{\kappa }\) from Brunner/Howard [1992].
Content:
Notes on \(UT(WO,\kappa ,WO),\ C(WO,\kappa )\) and \(D_{\kappa }\) from Brunner/Howard [1992].
Definition: \(D_{\kappa}\) or \(D(\kappa)\) is the sentence``Every non-well orderable set is the union of a disjoint, well orderable family of \(\kappa\) element sets.
(Note that \(UT(WO,\kappa ,WO)\),\(C(WO,\kappa)\), and \(D(\kappa)\) all depend on the well ordered, infinite cardinal \(\kappa\) and that \(UT(WO,<\aleph_{0},WO)\), \(UT(WO,<\aleph_{1},WO)\), and \(D(\aleph_{0})\) are forms [122 A], Form 151 and Form 152 respectively.)
Theorem: \((ZF^{0})\) \(UT(WO,\kappa ,WO)\) implies \(C(WO,\kappa )\) but does not imply \(C(\aleph_{0},\kappa^{+})\) nor does it imply \(D(\kappa^{+})\).
Theorem: \((ZF^{0})\) \(D(\kappa^+)\) does not imply \(D(\kappa)\).
Howard-Rubin number: 27
Type: Definitions and summaries
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