S. Shapiro, Epistemic and intuitionistic arithmetic, Intensional mathematics (S. Shapiro, editor), North-Holland, 1985, pp. 11--46.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....incompleteness theorem. In [26] cf. 59] Godel again acknowledged the problem of the provability semantics for S4. This issue was also addressed by Lemmon [44] Myhill [55] 56] Kripke [40] Montague [54] Mints [52] Kuznetsov Muravitsky [43] Goldblatt [27] Boolos [12] 14] Shapiro [62] [63], Buss [17] Artemov [1] and many others. However, the problem of finding an adequate provability semantics for S4 has remained open. A principal difficulty here is caused by the existential quantifier over proofs in the provability formula Provable(y) which is 9xProof (x; y) where Proof (x; y) ....

S. Shapiro. "Epistemic and Intuitionistic Arithmetic". In: S. Shapiro, ed., "Intensional mathematics", North-Holland, pp. 11-46, 1985.

....PA is provable in PA, which contradicts the Second Godel Incompleteness Theorem. The issue of a provability model for S4 was studied by Godel [18] Lemmon [29] Myhill [39] 40] Kripke [26] Montague [38] Mints [35] Kuznetsov Muravitskii [28] Goldblatt [19] Boolos [9] 10] Shapiro [43] [44], Buss [12] Artemov [1] and many others. However, the problem of a formal provability semantics for S4 has remained open. A principal difficulty here is caused by the existential quantifier over proofs in Provable(F ) Indeed, the interpretation of the formula 2(2F F ) is it is provable that ....

S. Shapiro. "Epistemic and Intuitionistic Arithmetic". In: S. Shapiro, ed., "Intensional mathematics", North-Holland, pp. 11-46, 1985. 58

....Interesting discussions of realizability, intuitionism, knowledge, and di# culties in the application of [3] to arithmetic, were given by Nelson in [26] and [27] These have guided our reasoning. Nelson also applied predicative concepts to define a constructive arithmetic in [25] Shapiro [28] used modal logic to define an epistemic arithmetic EA. He proved that it is as strong as HA under a Godel style mapping, while Goodman [15] used an infinitary cut elimination argument to show its conservativity over HA under a similar mapping. In contrast, we use Friedman s A translation to ....

S. Shapiro, "Epistemic and Intuitionistic Arithmetic", in Intensional Mathematics, S. Shapiro, ed., Studies in Logic and The Foundations of Mathematics v. 113, North-Holland, 1985.

....PA is provable in PA, which contradicts the Second Godel Incompleteness Theorem. The issue of a provability model for S4 was studied by Godel [13] Lemmon [20] Myhill [27] 28] Kripke [18] Montague [26] Mints [25] Kuznetsov Muravitskii [19] Goldblatt [14] Boolos [7] 8] Shapiro [30] [31], Buss [9] Artemov [1] and many others. However, the problem of a formal provability semantics for S4 has remained open. A principal difficulty here is caused by the existential quantifier over proofs in Provable(F ) Indeed, the interpretation of the formula 2(2F F ) is it is provable that ....

S. Shapiro. "Epistemic and Intuitionistic Arithmetic". In: S. Shapiro, ed., "Intensional mathematics", North-Holland, pp. 11-46, 1985.

....PA is provable in PA, which contradicts the second Godel incompleteness theorem. The issue of a provability model for S4 was studied by Godel [16] Lemmon [28] Myhill [35] 36] Kripke [25] Montague [34] Mints [33] Kuznetsov Muravitskii [27] Goldblatt [17] Boolos [9] 10] Shapiro [38] [39], Buss [11] Artemov [1] and many others. However, the problem of a formal provability semantics for S4 has remained open. A principal difficulty here is caused by the existential quantifier over proofs in Provable(F ) Indeed, the interpretation of the formula 2(2F F ) is it is provable that ....

S. Shapiro. "Epistemic and Intuitionistic Arithmetic". In: S. Shapiro, ed., "Intensional mathematics", North-Holland, pp. 11-46, 1985.

....of the form, 8 x 2 K; Kf( x) that is, 8 x; i Kx i ) oe Kf( x) For lack of a better word I call the system synthetic , in respect of the synthetic (constructive, non analytic) reading of t = t . There already is such a thing as epistemic arithmetic; see Goodman [9] and Shapiro [20]. A constructibility predicate Kx was defined by Lifschitz in his contribution to Shapiro s volume [19] the semantic meaning is given by a form of realizability. A special predicate letter is used to delimit the numbers in Peano s original formulation. Leivant extends this to a series of special ....

S. Shapiro, "Epistemic and Intuitionistic Arithmetic", in [19].

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S. Shapiro, Epistemic and intuitionistic arithmetic, Intensional mathematics (S. Shapiro, editor), North-Holland, 1985, pp. 11--46.

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