The Anatomy and Ultimate Nature of “True-But-Unprovable” Propositions

The class of all complementary pairs B, ∼B of propositions of CLFO [ = Classical Logic of Finite Order] is partitioned as {{U}, {K}, {N}}, where both {U} and {K} are recursively-enumerable. This leaves {N} as non r.e. and hence necessarily the “home” of the class of all “True-but-Unprovable” propositions B [together with their negations ∼B]. These “True-but-Unprovable” propositions B are true by virtue of being n-valid [ = valid on all-and-only-finite domains] and unprovable because ∼B has an infinite model. These B's are further characterized by containing some “item” of extra-logical [empirical] information, often of an existential nature, which comprises a condition which is incapable of being determined by purely logical means. Hence the truth of these B's is upon the truth of that “item” of extra-logical [empirical] information, and this fact is “witnessed” by the model of an alternate universe in which the contingency is false, and which is characterized by ∼B. A concrete example, Cantor's Theorem, is given, and the counter-model is constructed via a predicative definition.None of this detracts, or is intended to detract, from the elegant beauty and great importance of G\”odel's magnificent work [5], [6], [7]. Prior to G\”odel, no one except L\”owenheim [10] seems to have even of n-valid cwffs. G\”odel's incompleteness theorems force us to recognize the difference between truth and merely truth