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ZFC proves that the class of ordinals is not weakly compact for definable classes

Abstract:

In ZFC, the class Ord of ordinals is easily seen to satisfy the definable version of strong inaccessibility. Here we explore deeper ZFC-verifiable combinatorial properties of Ord, as indicated in Theorems A & B below. Note that Theorem A shows the unexpected result that Ord is never definably weakly compact in any model of ZFC.
Theorem A. Let be any model of ZFC.
(1)The definable tree property fails in : There is an -definable Ord-tree with no -definable cofinal branch.... Expand abstract
Publication status:
Published
Peer review status:
Peer reviewed

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Publisher copy:
10.1017/jsl.2017.75

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Institution:
University of Oxford
Division:
Humanities Division
Department:
Philosophy
Oxford college:
University College
Role:
Author
Publisher:
Association for Symbolic Logic Publisher's website
Journal:
Journal of Symbolic Logic Journal website
Volume:
83
Issue:
1
Pages:
146-164
Publication date:
2018-03-01
DOI:
EISSN:
1943-5886
ISSN:
0022-4812
Source identifiers:
916651
Language:
English
Keywords:
Pubs id:
pubs:916651
UUID:
uuid:746128df-b30b-4178-b52e-670dcc0adfde
Local pid:
pubs:916651
Deposit date:
2019-12-18

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