[Paper Review] Wetzel's Problem, Paul Erdos, and the Continuum Hypothesis: a mathematical mystery

Stephan Ramon Garcia, Amy L. Shoemaker|arXiv (Cornell University)|Jun 19, 2014

Quantum Mechanics and Applications2 references3 citations

TL;DR

This paper investigates the historical and mathematical origins of Wetzel's problem—whether a family of analytic functions with countable pointwise values must itself be countable—revealing that Paul Erdős proved the problem's answer is equivalent to the negation of the Continuum Hypothesis. The study traces the problem’s transmission from John E. Wetzel’s dissertation at Stanford to the University of Illinois Boneyard Book, where Lee A. Rubel likely recorded it, and ultimately to Erdős, who resolved it using deep set-theoretic methods, rendering the problem independent of ZFC.

ABSTRACT

This is a short historical note concerning the evolution of Wetzel's problem and Erdos' solution.

Motivation & Objective

• To reconstruct the historical provenance of Wetzel’s problem, from its inception in John E. Wetzel’s dissertation on harmonic functions to its transmission through academic networks.
• To clarify the source of the problem as cited by Paul Erdős in his 1963 paper, resolving the discrepancy between Erdős’ reference to the 'Ann Arbor Problem Book' and the actual location of the problem in the University of Illinois Boneyard Book.
• To establish that Erdős’ solution, which equates an affirmative answer to Wetzel’s problem with the negation of the Continuum Hypothesis, was based on a problem first recorded in the Boneyard Book by Lee A. Rubel.
• To resolve the mystery of how Erdős encountered the problem, identifying Lee A. Rubel as the most likely scribe and transmitter of the problem from Urbana-Champaign to Ann Arbor.
• To demonstrate that Wetzel’s problem is undecidable in ZFC, as it is logically equivalent to the negation of the Continuum Hypothesis.

Proposed method

• Historical archival research using university archives, personal correspondence, and digitized records from the Bentley Historical Library at the University of Michigan.
• Handwriting analysis comparing samples from Lee A. Rubel and entries in the Boneyard Book to identify the scribe of the original Wetzel problem entry.
• Cross-referencing of publication dates, conference records, and visit logs to reconstruct the timeline of Erdős’ visit to Ann Arbor in September 1963.
• Analysis of the content and structure of the Boneyard Book entries, including Bob Blakley’s and Bob Dixon’s contributions, to trace the evolution of the problem’s formulation.
• Use of mathematical logic to interpret Erdős’ proof, showing that the countability of function families under pointwise countability is equivalent to the negation of the Continuum Hypothesis.
• Corroboration of testimonies from mathematicians such as Peter Duren, John P. D’Angelo, and Wetzel himself to validate the reconstructed narrative.

Experimental results

Research questions

• RQ1 How did Wetzel’s original question about harmonic functions on Riemann surfaces evolve into a problem about analytic functions?
• RQ2 What was the actual provenance of the problem cited by Erdős as originating in the 'Ann Arbor Problem Book'?
• RQ3 Who was responsible for recording Wetzel’s problem in the Boneyard Book at the University of Illinois?
• RQ4 How did Paul Erdős come to solve the problem, and what role did the Boneyard Book play in his discovery?
• RQ5 Why did Erdős cite the 'Ann Arbor Problem Book' when the problem was actually recorded in the Boneyard Book at UIUC?

Key findings

• The problem originally posed by John E. Wetzel concerned harmonic functions on Riemann surfaces, not analytic functions, and was formulated during his dissertation work at Stanford in 1961.
• Lee A. Rubel is identified as the most likely author of the first recorded entry of Wetzel’s problem in the University of Illinois Boneyard Book, based on handwriting analysis and historical context.
• Paul Erdős encountered the problem in Ann Arbor during a visit in September 1963, likely from a copy of the problem circulated from the Boneyard Book, not from the lost Math Club book.
• Erdős proved that an affirmative answer to Wetzel’s problem is logically equivalent to the negation of the Continuum Hypothesis, rendering the problem independent of ZFC.
• Bob Dixon independently obtained a partial result assuming the Continuum Hypothesis, which was later superseded by Erdős’ full solution.
• The final version of Erdős’ paper was submitted to the Michigan Mathematical Journal on September 18, 1963, shortly after his visit to Ann Arbor, suggesting the problem was already known to him during that trip.