University of Gothenburg
In 1975 Barwise and Schlipf published a landmark paper which launched the study of recursively saturated models of arithmetic; the main theorem of their paper asserts that a nonstandard model M of Peano Arithmetic is recursively saturated iff M can be expanded to a model of a particular subsystem of Second Order Arithmetic. The impression one gets from reading the Barwise-Schlipf paper is that the left-to-right direction of the theorem is deep since it relies on sophisticated techniques from admissible set theory and infinitary languages, and that the other direction is fairly routine. In hindsight, we can say that the exact opposite is the case: the left-to-right direction of the theorem lends itself to a relatively easy proof from first principles, but the right-to-left direction is the deep one. In this talk I will provide some background history, discuss a serious gap in the original 1975-proof of the right-to-left direction, and explain how the gap can be circumvented using a line of reasoning very different from the original one. This talk is a report of joint work with Jim Schmerl.