Consistency and Konsistenz

Abstract

A ground-motive for this study of some historical and metaphysical implications of the diagonal lemmas of Cantor and Godel is Cantor's insightful remark to Dedekind in 1899 that the Inbegriff alles Denkbaren (aggregate of everything thinkable) might, like some class-theoretic entities, be “inkonsistent”. In the essay's opening sections, I trace some recent antecedents of Cantor's observation in logical writings of Bolzano and Dedekind (more remote counterparts of his language appear in the First Critique), then attempt to relativize the notion of Inkonsistenz to ‘self-sufficient’ theories T which interpret ⌈themselves⌉. In effect, I argue that Godel's diagonal lemma suggests a sense in which metatheoretic notions of proof, well-foundeness and satisfaction are object-theoretically inkonsistent. With respect to Cantor's Inbegriff, for example, the lemma yields that any object-theoretic ‘reconstruction’ θ of ‘thinkability’ generates an ‘antidiagonal’ sentence τ, which one can paraphrase as ⌈Self-referential application of the assertion that ⌈self-reference is unthinkable⌉⌉ is unthinkable. This sentence is provably equivalent to ⌈its own⌉ ‘unthinkability’. In the essay's last sections, I offer a skeptical interpretation of the metatheoretical consistency, for any consistent and ‘self-sufficient’ scientific or mathematical theory T, of the theory T+⌝Con(T). In effect, this amounts to a preliminary study of one form of malign-genius-argument for such theories T: what should a miniature ‘metaphysical realist’ c in a model M of T+⌝Con(T) make of an encounter (perhaps indirect) with one of M's witnesses [‘nonstandard’ to ‘us’] of ⌝Con(T)? In response to this query, I argue The thought-experiment thus seems to yield a genuine metatheoretic counterpart of Duhem's traditional problem of the boundary conditions of object-theoretic consistency: ‘we’ cannot ‘know’ we are not c.