Isomorphic limit ultrapowers for infinitary logic

The logic L θ 1 was introduced in [She12]; it is the maximal logic below L θ , θ 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 natur...

Full description

Saved in:
Bibliographic Details
Published in:Israel journal of mathematics 2021-12, Vol.246 (1), p.21-46
Main Author: Shelah, Saharon
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Group Theory and Generalizations
Interpolation
Logic
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The logic L θ 1 was introduced in [She12]; it is the maximal logic below L θ , θ 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 L θ 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 ℵ 0 , every complete L θ 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.
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-021-2226-x