In
mathematics, Wetzel's problem concerns bounds on the
cardinality of a set of
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s 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
The University of Illinois Urbana-Champaign (U of I, Illinois, University of Illinois, or UIUC) is a public land-grant research university in Illinois in the twin cities of Champaign and Urbana. It is the flagship institution of the Univer ...
.
[.][.]
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.
Paul Erdős in turn learned about the problem at the
University of Michigan
, mottoeng = "Arts, Knowledge, Truth"
, former_names = Catholepistemiad, or University of Michigania (1817–1821)
, budget = $10.3 billion (2021)
, endowment = $17 billion (2021)As o ...
, likely via
Lee Albert Rubel.
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.
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
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
is false.
[.] That is, the existence of an uncountable set of functions that maps each 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.
It follows from the independence of the continuum hypothesis, proved in 1963 by
Paul Cohen
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was award ...
, that the answer to Wetzel's problem is independent of
ZFC set theory.
Erdős' proof is so short and elegant that it is considered to be one of the
Proofs from THE BOOK.
In the case that the continuum hypothesis is false, Erdős asked whether there is a family of analytic functions, with the cardinality of the continuum, such that each complex number has a smaller-than-continuum set of images. As Ashutosh Kumar and
Saharon Shelah later proved, both positive and negative answers to this question are consistent.
References
{{reflist
Functional analysis
Independence results
Analytic functions