Description: The order extension principle (Form 49) together with 'Every linear ordering has a cofinal well ordered subset' (Form 51) implies AC (Form 1).
Content:
This note contains an argument thatThe order extension principle (Form 49) together with 'Every linear ordering has a cofinal well ordered subset (Form 51) implies AC (Form 1). This result was announced in Morris [1969].
Assume that \(Y\) is a set of non-empty sets. Let \(P\) be the set of choice functions \(f\) such that the domain of \(f\) is a subset of \(Y\). We define the following partial ordering on \(P\): \[ f \preceq_0 g \Leftrightarrow \mathrm{dom}(f) \subsetneq \mathrm{dom}(g) \lor f = g. \] Extend \(\preceq_0\) to a linear ordering \(\preceq\) on \(P\). Let \(\{f_\alpha: \alpha \in \beta\}\) be a cofinal subset of the linear ordering \((P,\preceq)\) where \(f_\alpha \prec f_\gamma \leftrightarrow \alpha \in \gamma\). (\(\in\) is the usual well ordering on the ordinals.) The set \(W = \bigcup_{\alpha\in\beta} \mathrm{dom}(f_\alpha)\) has a choice function \(F\) defined by \(F(x) = f_\alpha(x)\) where \(\alpha\) is the least ordinal for which \(x\in \mathrm{dom}(f_\alpha)\). We complete the proof by showing that \(W = Y\). If not, then there is some \(x \in Y-W\). Let \(t\) be any element of \(x\), then \(G = F\cup\{(x,t)\}\) is in \(P\) and has the property that \(\forall \alpha \in\beta\), \(f_\alpha \prec_0 G\) and hence, since \(\preceq\) extends \(\preceq_0\), \(f_\alpha\prec G\). This contradicts the cofinality of \(\{f_\alpha : \alpha \in \beta\}\).
Howard-Rubin number: 121
Type: proof of equivalencies
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