Consistency and konsistenz

Erkenntnis 26 (1):1 - 43 (1987)
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Abstract

A ground-motive for this study of some historical and metaphysical implications of the diagonal lemmas of Cantor and Gödel 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 Gödel'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 of thinkability generates an antidiagonal sentence , which one can paraphrase asSelf-referential application of the assertion that.

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Citations of this work

Set theory and physics.K. Svozil - 1995 - Foundations of Physics 25 (11):1541-1560.

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References found in this work

Completeness in the theory of types.Leon Henkin - 1950 - Journal of Symbolic Logic 15 (2):81-91.
Paradoxien des Unendlichen.Bernard Bolzano - 2012 - Hamburg: Meiner, F. Edited by Christian Tapp.
Logic in the twenties: The nature of the quantifier.Warren D. Goldfarb - 1979 - Journal of Symbolic Logic 44 (3):351-368.
The incompleteness theorems.Craig Smorynski - 1977 - In Jon Barwise (ed.), Handbook of Mathematical Logic. North-Holland. pp. 821 -- 865.

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