Erdős and Set Theory

Paul Erdős (26 March 1913 20 September 1996) was a mathematician par excellence whose results and initiatives have had a large impact and made a strong imprint on the doing of and thinking about mathematics. A mathematician of alacrity, detail, and collaboration, Erdős in his six decades of work moved and thought quickly, entertained increasingly many parameters, and wrote over 1500 articles, the majority with others. His modus operandi was to drive mathematics through cycles of problem, proof, and conjecture, ceaselessly progressing and ever reaching, and his modus vivendi was to be itinerant in the world, stimulating and interacting about mathematics at every port and capital. Erdős’ main mathematical incentives were to count, to estimate, to bound, to interpolate, and to get at the extremal or delimiting, and his main emphases were on elementary and random methods. These had a broad reach across mathematics but was particularly synergistic with the fields that Erdős worked in and developed. His mainstays were formerly additive and multiplicative number theory and latterly combinatorics and graph theory, but he ranged across and brought in probability and ergodic theory, the constructive theory of functions and series, combinatorial geometry, and set theory. He had a principal role in establishing probabilistic number theory, extremal combinatorics, random graphs, and the partition calculus for infinite cardinals. Against this backdrop, this article provides an account of Erdős’ work and initiatives in set theory with stress put on their impact on the subject. Erdős importantly contributed to set theory as it became a broad, sophisticated field of mathematics in two dynamic ways. In the early years, he established results and pressed themes that would figure pivotally in formative advances. Later and throughout, he followed up on combinatorial initiatives that became part and parcel of set theory. Emergent from combinatorial thinking, Erdős’ results and initiatives in set theory had the feel of being simple and basic yet rich and pivotal, and so accrued into the subject as seminal at first, then formative, and finally central. Proceeding chronologically, we work to draw all this out as well as make connections with Erdős’ larger work and thinking, to bring out how it is all of a piece. Paul Erdős and His Mathematics, in two volumes [Halász et al., 2002a] and [Halász et al., 2002b], emanated from a 1999 celebratory conference, and it surveys Erdős’ work, provides reminiscences, and contains research articles. The Mathematics of Paul Erdős, in two volumes [Graham et al., 2013a] and