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

Journal article
Authors Ali Enayat
Joel David Hamkins
Published in Journal of Symbolic Logic
Volume 83
Issue 1
Pages 146-164
ISSN 00224812
Publication year 2018
Published at Department of Philosophy, Linguistics and Theory of Science
Pages 146-164
Language en
Keywords Gödel-Bernays class theory, weakly compact cardinal, Zermelo-Fraenkel set theory
Subject categories Logic, Algebra and Logic


© 2018 The Association for Symbolic Logic. 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. LetMbe any model of ZFC. (1) The definable tree property fails in M: There is an M-definable Ord-tree with no M-definable cofinal branch. (2) The definable partition property fails in M: There is an M-definable 2-coloring f: [X] 2 → 2 for someM-definable proper class X such that no M-definable proper classs is monochromatic for f. (3) The definable compactness property for L∞, ω fails in M: There is a definable theory Γ in the logic L∞,ω (in the sense ofM) of size Ord such that every set-sized subtheory of Γ is satisfiable in M, but there is no M-definable model of Γ. Theorem B. The definable Ord principle holds in a model M of ZFC iff M carries an M-definable global well-ordering. Theorems A and Babove can be recast as theoremschemes in ZFC, or as asserting that a single statement in the language of class theory holds in all 'spartan' models of GB (Godel-Bernays class theory); where a spartan model of GB is any structure of the form (M,DM), where M | = ZF and DM is the family of M-definable classes. Theorem C gauges the complexity of the collection GB spa of (Godel-numbers of) sentences that hold in a ll spartan models of GB. Theorem C. GB spa is Π 1 1 -complete.

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