## Expansions omitting a type

## R. Kaye (logic seminar 29.11.2006)

Abstract:

A model is resplendent if for any recursively axiomatized theory T in an expanded language which is consistent with the theory of the model there is an expansion of the model satisfying T. For countable models in recursive languages, this notion corresponds to recursive saturation, and resplendent models exist in all cardinalities. The notion arises from work in the 1970s by Barwise, Schlipf and Ressayre and has been much used in models of arithmetic. Comparitively recently, the strengthening of recursive saturation to arithmetical saturation has proved a useful notion, though "arithmetical saturation" is not known to correspond to any particularly elegant strengthening of "resplendency". We propose the study of a notion of resplendency where as well as satisfying a theory T, a type p is omitted in the expansion. This new version of resplendency, called "transplendency" will be defined and studied, and transplendent models shown to exist. Some properties of transplendent models, espacially transplendent models of arithmetic, will be discussed and it will be shown that tranplendency is much stronger than arithmetical saturation. The exact strength of transplendency is unknown, and seems to depend on some issues in descriptive set theory.

(Joint work with Fredrik Engstrom, Sweden)