In the 60's, Erdős showed that the continuum hypothesis is equivalent to the statement that there is an uncountable family of entire functions on the complex plane that attains only countably many values at each point. The argument in fact shows that any family of entire functions, that attains at each point less values than elements of that family, must have size continuum. Recently Kumar and Shelah have shown that consistently such a family exists while the continuum has size \(\aleph_{\omega_1}\). We answer their main open problem by showing that continuum \(\aleph_2\) is possible as well.
This is joint work with T. Weinert.
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