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

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.