Isomorphic limit ultrapowers for infinitary logic

Abstract

The logic \({\mathbb{L}}_\theta ^1\) was introduced in [She12]; it is the maximal logic below \({{\mathbb{L}}_{\theta, \theta}}\) in which a well ordering is not definable. We investigate it for θ a compact cardinal. We prove that it satisfies several parallels of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, two models are \({\mathbb{L}}_\theta ^1\)-equivalent iff for some ω-sequence of θ-complete ultrafilters, the iterated ultrapowers by it of those two models are isomorphic.Also for strong limit λ>θ of cofinality \({\aleph _0}\), every complete \({\mathbb{L}}_\theta ^1\)-theory has a so-called special model of cardinality λ, a parallel of saturated. For first order theory T and singular strong limit cardinal λ, T has a so-called special model of cardinality λ. Using “special” in our context is justified by: it is unique (fixing T and λ), all reducts of a special model are special too, so we have another proof of interpolation in this case.