Abstract
We present a semantic proof of Löb's theorem for theories T containing ZF. Without using the diagonalization lemma, we construct a sentence AUT T, which says intuitively that the predicate “autological with respect to T” (i.e. “applying to itself in every model of T”) is itself autological with respect to T. In effect, the sentence AUT T states “I follow semantically from T”. Then we show that this sentence indeed follows from T and therefore is true.
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References
Adamowicz, Z., and T. Bigorajska ‘Existentially closed structures and Gödel's second incompleteness theorem’, Journal of Symbolic Logic, 66(1) (2001).
Barwise, J., and J. Etchemendy, The Liar, Oxford University Press, New York, 1987.
Van Benthem, J., ‘Four paradoxes’, Journal of Philosophical Logic 7 (1978).
CieŚliŃski, C., ‘Heterologicality and incompleteness’, Mathematical Logic Quarterly 48 (2002).
Curry, H., ‘The inconsistency of certain formal logics’, Journal of Symbolic Logic 7 (1942).
Krivine, J.-L., Théorie Axiomatique des Ensembles, PUF, Paris 1972.
Kunen, K., Set Theory. An Introduction to Independence Proofs, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1980.
LÖb, M. H., ‘Solution to a problem of Leon Henkin’, Journal of Symbolic Logic 40 (1955), 115–118.
Smorynski, C., ‘The Incompleteness Theorems’, in J. Barwise, (ed.), Handbook of Mathematical Logic, North-Holland Publishing Co., Amsterdam, 1977.
Smorynski, C., ‘The development of self-reference: Löb's theorem’, in T. Drucker, (ed.), Perspectives on the History of Mathematical Logic, Birkhäuser 1991, pp. 111–133.
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Cieśliński, C. Löb's Theorem in a Set Theoretical Setting. Studia Logica 75, 319–326 (2003). https://doi.org/10.1023/B:STUD.0000009563.84627.cd
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DOI: https://doi.org/10.1023/B:STUD.0000009563.84627.cd