mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Wetzel's problem concerns bounds on the

cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...

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 (UIUC, U of I, Illinois, or University of Illinois) is a public land-grant research university in the Champaign–Urbana metropolitan area, Illinois, United States. Established in 1867, it is the f ...

... 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 In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...

of values. In his doctoral dissertation, Wetzel asked whether this assumption implies that ''F'' is necessarily itself countable.

Paul Erdős Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, g ...

in turn learned about the problem at the

University of Michigan The University of Michigan (U-M, U of M, or Michigan) is a public university, public research university in Ann Arbor, Michigan, United States. Founded in 1817, it is the oldest institution of higher education in the state. The University of Mi ...

, 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, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...

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, 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 awarded a F ...

, 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 ''Proofs from THE BOOK'' is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathemat ...

. 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 Saharon Shelah (; , ; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey. Biography Shelah was born in Jerusalem on July 3, 1945. He is th ...

later proved, both positive and negative answers to this question are consistent.

References

{{reflist Functional analysis Independence results Analytic functions