Description:
Form 289 (If \(S\) is a set of subsets of acountable set and \(S\) is closed under chain unions, then \(S\) has a \(\subseteq\)-maximal element.) is false in \(\cal M1\).
Content:
Form 289 (If \(S\) is a set of subsets of acountable set and \(S\) is closed under chain unions, then \(S\) has a \(\subseteq\)-maximal element.) is false in \(\cal M1\).
Let \(b\) be the generic set of generic reals in \(\cal M1\), let \(E(b)\) be the set of all finite subsets of \(b\), and let \(S = \{\cup x : x\in E(b)\}\). Since \(b\) is generic, for any finite subset \(\{ a_0, a_1, \ldots, a_n\}\) of \(b\), \(a_0 \not\subseteq \cup \{a_1, \ldots, a_n\}\) It follows that if\(x\), \(y\in E(b)\), then \(\cup x\subseteq \cup y\) iff \(x\subseteq y\). Hence, any chain in \(S\) corresponds to a chain in \(E(b)\) and a \(\subseteq\)-maximal element of \(S\) corresponds to a \(\subseteq\)-maximal element of \(E(b)\). We claim that there is no infinite chain in \(E(b)\). For suppose\(x_0\subset x_1\subset x_2 \cdots\) is an infinite chain in \(E(b)\). Let \(a_n\) be the smallest element in \(x_{n+1}-x_n\) for \(n\in\omega\). Then \(\{a_n : n\in\omega\}\) is a denumerable subset of \(b\), which is impossible because \(b\) is Dedekind finite. Thus, all chains in \(E(b)\) are finite so \(E(b)\) is closed under chain unions. Consequently, \(S\) is closed under chain unions. Since \(b\) is infinite, \(E(b)\) has no \(\subseteq\)-maximal element and, therefore, \(S\) has no \(\subseteq\)-maximal element.
Howard-Rubin number:
102
Type:
Theorem
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