Details

Autor(en) / Beteiligte

• ENAYAT, ALI
• HAMKINS, JOEL DAVID

Titel

ZFC PROVES THAT THE CLASS OF ORDINALS IS NOT WEAKLY COMPACT FOR DEFINABLE CLASSES

Ist Teil von

• The Journal of symbolic logic, 2018-03, Vol.83 (1), p.146-164

Ort / Verlag

Pasadena: Association for Symbolic Logic, Inc

Erscheinungsjahr

2018

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• 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 M be 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 some M-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 of M) 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 B above can be recast as theorem schemes in ZFC, or as asserting that a single statement in the language of class theory holds in all ‘spartan’ models of GB (Gödel-Bernays class theory); where a spartan model of GB is any structure of the form (M, D M), where M ⊨ ZF and D M is the family of M-definable classes. Theorem C gauges the complexity of the collection GBspa of (Gödel-numbers of) sentences that hold in all spartan models of GB. Theorem C. GBspa is Π 1 1 -complete.