Form equivalence class Howard-Rubin Number: 14
Statement: Suppose to each finite subset \(F\) of a set I therecorresponds a set \(\Phi (F)\) of functions whose domains are subsetsof \(I\) including \(F\) and such that (a) \(F_{1}\subseteq F_{2}\) implies\(\Phi(F_{1})\subseteq\Phi(F_{2})\) and (b) \(\forall i\in I\), \(\bigl\{\phi(i): \phi\in\bigcup\{\Phi(F): F\hbox{ finite and }F\subseteq I\}\bigr\}\)is finite. Then there is a function \(f\) with domain \(I\) suchthat for all finite \(F\subseteq I\), \(\exists\phi\in\Phi(F)\)such that \(\phi\) and \(f\) coincide on F. Rav [1977], Cowen [1973] and Note 80.
Howard-Rubin number: 14 AP
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