Wetzel's Problem and the Continuum Hypothesis

 

Title: Wetzel's Problem and the Continuum Hypothesis
Author: Lawrence C Paulson
Submission date: 2022-02-18
Abstract: Let $F$ be a set of analytic functions on the complex plane such that, for each $z\in\mathbb{C}$, the set $\{f(z) \mid f\in F\}$ is countable; must then $F$ itself be countable? The answer is yes if the Continuum Hypothesis is false, i.e., if the cardinality of $\mathbb{R}$ exceeds $\aleph_1$. But if CH is true then such an $F$, of cardinality $\aleph_1$, can be constructed by transfinite recursion. The formal proof illustrates reasoning about complex analysis (analytic and homomorphic functions) and set theory (transfinite cardinalities) in a single setting. The mathematical text comes from Proofs from THE BOOK by Aigner and Ziegler.
BibTeX:
@article{Wetzels_Problem-AFP,
  author  = {Lawrence C Paulson},
  title   = {Wetzel's Problem and the Continuum Hypothesis},
  journal = {Archive of Formal Proofs},
  month   = feb,
  year    = 2022,
  note    = {\url{https://isa-afp.org/entries/Wetzels_Problem.html},
            Formal proof development},
  ISSN    = {2150-914x},
}
License: BSD License
Depends on: ZFC_in_HOL