Description: In this note we prove that \(C(\infty, \aleph_0)\) (Form 62) and that \(MC(\infty, \aleph_0)\) (Form 349) are false in Howard's Model III (\(\mathcal{N}\)56) and that \(C(WO, \infty)\) (Form 40) is true.
Content:
\(C(\infty, \aleph_0)\) is false because the set \(\mathfrak{A} = \{ A_i : i \in \mathbb{Q} \}\) is in the model and each \(A_i\) is countable in the model but \(\mathfrak{A}\) has no choice function in the model.
Similarly \(\mathfrak{A}\) has no multiple choice function in the model so \(MC(\infty, \aleph_0)\) is false.
For the proof that \(C(WO, \infty)\) is true, assume that \(\mathcal{X}\) is a well ordered collection of non-empty sets which is in the model along with a well ordering (of \(\mathcal{X}\)). Then there is some bounded subset \(E\) of \(\mathbb{Q}\), which we assume without loss of generality is an interval \((a,b)\), such that \(\bigcup_{i \in E} A_i\) is a support of \(\mathcal{X} \) and every element of \(\mathcal{X}\). Choose rational numbers \(a'\) and \(b'\) such that \(a' < a\) and \(b < b'\) and let \(E' = (a',b')\). We claim that \(\bigcup_{i \in E} A_i'\) is a support for a choice function on \(X\). In order to prove this it suffices to show that \(\forall X \in \mathcal{X}, \exists y \in X\) such that \(\bigcup_{i \in E'} A_i\) is a support of \(y\).
Assume \(X \in \mathcal{X}\), that \(x \in \mathbb{X}\) and that \(x\) has support \(\bigcup_{i \in E_x} A_i\) where \(E_x\) is a bounded subset of \(\mathbb{Q}\). Then there is a permutation \(\phi\) of \(\mathbb{Q}\) such that \(\phi\) fixes \((a,b)\) pointwise and such that \(\phi(E_x) \subset E'\). Let \(\phi^*\) be the element of \(\mathcal{G}\) defined by \(\phi^*(a(i,j)) = a(\phi(i), j) \). Then \(\phi^*\) fixes \(\bigcup_{i \in E} A_i\) pointwise and therefore fixes \(X\). So \(\phi^*(x) \in X\). Further \(\phi^*(\bigcup_{i \in E_x} A_i)\) is a support of \(\phi^*(x)\). Since \(\phi^*(\bigcup_{i \in E_x} A_i) \subseteq \bigcup_{i \in E'} A_i\) the element \(y = \phi^*(x)\) satisfies the required conditions.
Howard-Rubin number: 163
Type: proofs of result
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