Author:
ENAYAT ALI,HAMKINS JOEL DAVID
Abstract
AbstractIn 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${\cal M}$be any model of ZFC.(1)The definable tree property fails in${\cal M}$: There is an${\cal M}$-definable Ord-tree with no${\cal M}$-definable cofinal branch.(2)The definable partition property fails in${\cal M}$: There is an${\cal M}$-definable 2-coloring$f:{[X]^2} \to 2$for some${\cal M}$-definable proper class X such that no${\cal M}$-definable proper classs is monochromatic for f.(3)The definable compactness property for${{\cal L}_{\infty ,\omega }}$fails in${\cal M}$: There is a definable theory${\rm{\Gamma }}$in the logic${{\cal L}_{\infty ,\omega }}$(in the sense of${\cal M}$) of size Ord such that every set-sized subtheory of${\rm{\Gamma }}$is satisfiable in${\cal M}$, but there is no${\cal M}$-definable model of${\rm{\Gamma }}$.Theorem B. The definable ⋄Ordprinciple holds in a model${\cal M}$of ZFC iff${\cal M}$carries an${\cal M}$-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 $\left( {{\cal M},{D_{\cal M}}} \right)$, where ${\cal M} \models {\rm{ZF}}$ and ${D_{\cal M}}$ is the family of${\cal M}$-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. GBspais${\rm{\Pi }}_1^1$-complete.
Publisher
Cambridge University Press (CUP)
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