Expansions omitting a type
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)