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

<br/>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 &amp; B below. Note that Theorem A shows the unexpected result that Ord is never definably weakly compact in any model of ZFC. <br/>Theorem A. Let be any model of ZFC. <br/>(1)The definable tree property fails in : There is an -definable Ord-tree with no -definable cofinal branch. <br/>(2)The definable partition property fails in : There is an -definable 2-coloring for some -definable proper class X such that no -definable proper classs is monochromatic for f. <br/>(3)The definable compactness property for fails in : There is a definable theory in the logic (in the sense of ) of size Ord such that every set-sized subtheory of is satisfiable in , but there is no -definable model of . Theorem B. The definable ⋄Ord principle holds in a model of ZFC iff carries an -definable global well-ordering. <br/>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 , where and is the family of -definable classes. Theorem C gauges the complexity of the collection GBspa of (Gödel-numbers of) sentences that hold in all spartan models of GB. <br/>Theorem C. GBspa is -complete.