Form equivalence class Howard-Rubin Number: 0
Statement:
Suppose \(X\) and \(M\) are sets, \(F\) is a function from\({\cal P}(X) - \{\emptyset\}\) into \(M\) and \(G\) is a function on \(M\) such that \(\forall z\in {\cal P}(X) - \{\emptyset\}\), \(G(F(z))\in {\cal P}(z)- \{\emptyset\}\). Then there are unique \(W\) and \(R\) such that \(W \subseteq M\), \(R\subseteq W\times W\) and \(R\) is an irreflexive well ordering of \(W\),the range of \(G|W\) is a partition of \(X\), and \(\forall w\in W\), \(w =F(X -(\bigcup\{G(\nu): \nu\in W\) and \(\nu\mathrel R w\}))\).
Howard-Rubin number: 0 Z
Citations (articles):
Kruse [1974]
Some results on partitions and Cartesian products in the absence of the axiom of choice
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