mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Wetzel's problem concerns bounds on the

cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

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... 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 numbers; ...

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

Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 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 ...

in turn learned about the problem at the University of Michigan, likely via

Lee Albert Rubel Lee Albert Rubel ( – ) was a mathematician known for his contributions to analog computing. Career Originally from New York, he held a Doctorate of Mathematics degree from University of Wisconsin-Madison, and was professor of Mathematics at Un ...

. 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 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, 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 mat ...

. 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 ( he, שהרן שלח; 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, ...

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

References

{{reflist Functional analysis Independence results Analytic functions