Wetzel's problem

In mathematics, Wetzel's problem concerns bounds on the cardinality of a set of analytic functions that, for each of their arguments, take on few distinct values. It is named after John Wetzel, a mathematician at the University of Illinois at Urbana–Champaign.[1][2]

Let F be a family of distinct analytic functions on a given domain with the property that, for each x in the domain, the functions in F map x to a countable set of values. In his doctoral dissertation, Wetzel asked whether this assumption implies that F is necessarily itself countable.[3] Paul Erdős in turn learned about the problem at the University of Michigan, likely via Lee Albert Rubel.[1] In his paper on the problem, Erdős credited an anonymous mathematician with the observation that, when each x is mapped to a finite set of values, F is necessarily finite.[4]

However, as Erdős showed, the situation for countable sets is more complicated: the answer to Wetzel's question is yes if and only if the continuum hypothesis is false.[4] That is, the existence of an uncountable set of functions that maps any argument x to a countable set of values is equivalent to the nonexistence of an uncountable set of real numbers whose cardinality is less than the cardinality of the set of all real numbers. One direction of this equivalence was also proven independently, but not published, by another UIUC mathematician, Robert Dan Dixon.[1] It follows from the independence of the continuum hypothesis, proved in 1963 by Paul Cohen,[5] that the answer to Wetzel's problem is independent of ZFC set theory.[1]

References

  1. 1 2 3 4 Garcia, Stephan Ramon; Shoemaker, Amy L. (March 2015), "Wetzel's problem, Paul Erdős, and the continuum hypothesis: a mathematical mystery", Notices of the AMS 62 (3): 243–247, arXiv:1406.5085.
  2. ↑ Aigner, Martin; Ziegler, Günter M. (2014), Proofs from The Book (5th ed.), Springer-Verlag, Berlin, pp. 132–134, doi:10.1007/978-3-662-44205-0, ISBN 978-3-662-44204-3, MR 3288091.
  3. ↑ Wetzel, John Edward (1964), A Compactification Theory with Potential-Theoretic Applications, Ph.D. thesis, Stanford University, p. 98. As cited by Garcia & Shoemaker (2015).
  4. 1 2 Erdős, P. (1964), "An interpolation problem associated with the continuum hypothesis", The Michigan Mathematical Journal 11: 9–10, doi:10.1307/mmj/1028999028, MR 0168482.
  5. ↑ Cohen, Paul J. (December 15, 1963), "The Independence of the Continuum Hypothesis", Proceedings of the National Academy of Sciences of the United States of America 50 (6): 1143–1148, doi:10.1073/pnas.50.6.1143, JSTOR 71858, PMC 221287, PMID 16578557.
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