Implications

We have the following indirect implication of form equivalence classes:

184 \(\Rightarrow\) 93
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
184 \(\Rightarrow\) 93 Double uniformization, Kaniewski, J. 1980, J. London Math. Soc.

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
184:

Existence of a double uniformization: For all \(X\) and \(Y\), for all \(E\subseteq X\times Y\), if there is an infinite cardinal \(\kappa\) satisfying:

  1. \(\forall y_0\in Y\), \(|\{(x,y_0): x\in X\}\cap E| =\kappa\) and
  2. \(\forall x_0\in X\), \(|\{(x_0,y): y\in Y\}\cap E| = \kappa\),
then \(\exists U\subseteq E\) such that for all \((x,y)\in E, \exists !z\in X\) such that \((z,y)\in U\) and \(\exists !z\in Y\) such that \((x,z)\in U\). (\(U\) is called a double uniformization of \(E\).)

93:

There is a non-measurable subset of \({\Bbb R}\).

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