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In this note we prove that form [30 G] (Every \(\subseteq\)-chain in \(\cal P(X)\) is contained in a maximal \(\subseteq\)-chain) is equivalent to Form 30

Content:

I n this note we prove that form [30 G] (Every \(\subseteq\)-chain in \(\cal P(X)\) is contained in a maximal \(\subseteq\)-chain) is equivalent to Form 30 (The ordering principle). Form [30 G] was suggested by Adam Kolany. Assume [30 G].  Let \(X\) be a set and let \(Y\) be a maximal \(\subseteq\)-chain in \(\cal P(X)\).  Define \(\le\) on \(X\) by \(a \le b\) iff \((\exists A\in Y)(a\in A \land b\notin A)\). \(\le\) is linear because if not there are \(a,b\in X\) such that \(a\not\le b\) and \(b\not\le a\). If this happens then \(Y \cup \{A_0\}\) (where \(A_0 =(\bigcup\{ A\in Y : a\notin A \})\cup \{b\}\)) is a chain in \(\cal P(X)\) which properly extends \(Y\).
Assume Form 30.  Let \(X\) be a set and \(Y\) a \(\subseteq\)-chain in \(\cal P(X)\).  Let \(\le\) be a linear ordering of \(X\). Define another linear ordering \(R\) on \(X\) by \(a\mathrel R b\) iff \([(\exists A\in Y)(a\in A\land b\notin A)]\) or \([(\forall A\in Y)(a\in A \leftrightarrow b\in A)\land a\le b]\).  Then \(Y' = \{ A\subseteq X : (\forall a\in A)(\forall x \mathrel R a)(x\in A) \}\) is a maximal \(\subseteq\)-chain containing \(Y\).

Howard-Rubin number: 130

Type: Equivalences

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