Scott’s problem for models of set theory
This was my oral exam, which took the form of what was essentially a talk at the CUNY Set Theory Seminar.
Scott’s problem asks which Scott sets can be represented as standard systems of models of Peano arithmetic. Knight and Nadel proved in 1982 that every Scott set of cardinality is the standard system of some model of PA. Little progress has been made since then on the problem. I will present a modification of these results to models of ZFC and show that every Scott set of cardinality is the standard system of some model of ZFC. This talk will constitute the speaker’s oral exam.