## Achievements

Posted: Wednesday, September 7, 2022### Daniel Cunningham, Professor Emeritus, Mathematics

Daniel Cunningham, professor emeritus of mathematics, has just published his paper "On Forcing over L(R)" in the *Archive for Mathematical Logic *(AML). The paper is available online and will soon be assigned to an issue. The AML is a refereed journal that publishes research papers on mathematical logic and set theory.

In set theory, the method of forcing consists of extending a set theoretical universe M, called an inner model, to a larger universe M* that contains new sets that are not in the old universe M; however, there are forcing extensions that add too many new sets. In particular, virtually all nontrivial forcing extensions of the inner model L(R) do not satisfy the axiom of determinacy (AD), simply because such forcing adds too many new sets. A natural question to ask is, When can one use a forcing extension to add only a small number of new sets to L(R) and then obtain an inner model (of the extension) that satisfies AD? In his paper, Dr. Cunningham shows that this can be done and provides examples of such extensions.

Paul Cohen developed the forcing technique to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. For his work in set theory, Dr. Cohen received the Fields Medal, which is awarded every four years on the occasion of the International Congress of Mathematicians to recognize outstanding mathematical achievement. Ever since Dr. Cohen introduced the forcing concept to set theory and mathematics, forcing has been seen by the general mathematical community as a subject of great intrinsic interest, but one that is technically forbidding except to those whose specialty is in set theory. Books and papers have been published to alleviate this misperception.