Chain models, trees of singular cardinality and dynamic EF games

Džamonja, Mirna and Väänänen, Jouko (2011) Chain models, trees of singular cardinality and dynamic EF games. Journal of Mathematical Logic (jml), 11 (1). pp. 61-85. ISSN 0219-0613

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Abstract

Let κ be a singular cardinal. Karp's notion of a chain model of size ? is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf(κ). With a notion of satisfaction and (chain)-isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game is associated a tree which is a tree of size κ with no κ-branches (even no cf(κ)-branches). The measure of how non-isomorphic the models are is reflected by a certain order on these trees, called reduction. We study the collection of trees of size κ with no κ-branches under this notion and prove that when cf(κ) = ω this collection is rather regular; in particular it has universality number exactly κ+. Such trees are then used to develop a descriptive set theory of the space cf(κ)κ.The main result of the paper gives in the case of κ strong limit singular an exact connection between the descriptive set-theoretic complexity of the chain isomorphism orbit of a model, the reduction order on the trees and winning strategies in the corresponding dynamic EF games. In particular we obtain a neat analog of the notion of Scott watershed from the Scott analysis of countable models.

Item Type:	Article
Faculty \ School:	Faculty of Science > School of Mathematics
UEA Research Groups:	Faculty of Science > Research Groups > Logic
Depositing User:	Vishal Gautam
Date Deposited:	18 Mar 2011 10:19
Last Modified:	15 Dec 2022 01:23
URI:	https://ueaeprints.uea.ac.uk/id/eprint/19971
DOI:	10.1142/S0219061311001006

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