On Scott and Karp trees of uncountable models

Journal of Symbolic Logic 55 (3):897-908 (1990)
  Copy   BIBTEX

Abstract

Let U and B be two countable relational models of the same first order language. If the models are nonisomorphic, there is a unique countable ordinal α with the property that $\mathfrak{U} \equiv^\alpha_{\infty\omega} \mathfrak{B} \text{but not} \mathfrak{U} \equiv^{\alpha + 1}_{\infty\omega} \mathfrak{B},$ i.e. U and B are L ∞ω -equivalent up to quantifier-rank α but not up to α + 1. In this paper we consider models U and B of cardinality ω 1 and construct trees which have a similar relation to U and B as α above. For this purpose we introduce a new ordering T ≪ T' of trees, which may have some independent interest of its own. It turns out that the above ordinal α has two qualities which coincide in countable models but will differ in uncountable models. Respectively, two kinds of trees emerge from α. We call them Scott trees and Karp trees, respectively. The definition and existence of these trees is based on an examination of the Ehrenfeucht game of length ω 1 between U and B. We construct two models of power ω 1 with 2 ω 1 mutually noncomparable Scott trees

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 92,323

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Analytics

Added to PP
2009-01-28

Downloads
56 (#287,373)

6 months
18 (#144,494)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Jouko A Vaananen
University of Helsinki

Citations of this work

Closed Maximality Principles and Generalized Baire Spaces.Philipp Lücke - 2019 - Notre Dame Journal of Formal Logic 60 (2):253-282.
Game-theoretic inductive definability.Juha Oikkonen & Jouko Väänänen - 1993 - Annals of Pure and Applied Logic 65 (3):265-306.
Observations about Scott and Karp trees.Taneli Huuskonen - 1995 - Annals of Pure and Applied Logic 76 (3):201-230.

Add more citations

References found in this work

Set Theory.Keith J. Devlin - 1981 - Journal of Symbolic Logic 46 (4):876-877.
A new approach to infinitary languages.J. Hintikka - 1976 - Annals of Mathematical Logic 10 (1):95.
Countable approximations and Löwenheim-Skolem theorems.David W. Kueker - 1977 - Annals of Mathematical Logic 11 (1):57.

Add more references