Singular Cardinals: From Hausdorff's Gaps to Shelah's PCF Theory

The mathematical subject of singular cardinals is young and many of the mathematicians who made important contributions to it are still active. This makes writing a history of singular cardinals a somewhat riskier mission than writing the history of, say, Babylonian arithmetic. Yet exactly the discussions with some of the people who created the 20th century history of singular cardinals made the writing of this article fascinating. I am indebted to Moti Gitik, Ronald Jensen, István Juhász, Menachem Magidor and Saharon Shelah for the time and effort they spent on helping me understand the development of the subject and for many illuminations they provided. A lot of what I thought about the history of singular cardinals had to change as a result of these discussions. Special thanks are due to István Juhász, for his patient reading for me from the Russian text of Alexandrov and Urysohn's Memoirs, to Salma Kuhlmann, who directed me to the definition of singular cardinals in Hausdorff's writing, and to Stefan Geschke, who helped me with the German texts I needed to read and sometimes translate. I am also indebted to the Hausdorff project in Bonn, for publishing a beautiful annotated volume of Hausdorff's monumental Grundzüge der Mengenlehre and for Springer Verlag, for rushing to me a free copy of this book; many important details about the early history of the subject were drawn from this volume. The wonderful library and archive of the Institute Mittag-Leffler are a treasure for anyone interested in mathematics at the turn of the 20th century; a particularly pleasant duty for me is to thank the institute for hosting me during my visit in September of 2009, which allowed me to verify various details in the early research literature, as well as providing me the company of many set theorists and model theorists who are interested in the subject. I an grateful to all colleagues who read early versions of this manuscript and who made many valuable comments. Aki Kanamori, who spent too many hours of his time reading and correcting my writing deserves my deepest gratitude. Needless to say, all mistakes which remained in this text are solely mine.