Undecidability in diagonalizable algebras, Journal of Symbolic Logic, 62, pp. 79--116. C. Smory' nski Home/Search Document Not in Database Summary Related Articles Check |
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The Logic of Provability - Japaridze, de Jongh (1998) (13 citations) (Correct)
....not propositional theories in the above sense. Let us call a set T of modal formulas realistic if there exists a realization such that A is true, for every A 2 T . Moreover, we say that T is well specified if, whenever A 2 T and B is a subformula of A, we also have B 2 T or :B 2 T . Strannegard [1997] proves a result that generalizes both theorem 5.1 and Solovay s second arithmetic completeness theorem. We give a weak but easy to state version of it. 5.3. Theorem. Let T be a well specified r.e. set of modal formulas. Then T is realistic iff T is consistent with S. An even more general point ....
....concerning the Magari algebras of formal theories that cannot naturally be formulated in terms of propositional theories. 486 G. Japaridze and D. de Jongh 5.4. Theorem. Shavrukov [1993a] The Magari algebras of PA and ZF are not isomorphic, and, in fact not elementarily equivalent (Shavrukov [1997]) The proof only uses the fact that ZF proves the uniform Sigma 1 reflection principle for PA. A corollary of the theorem is that there is a formula of the second order propositional calculus that is valid in the interpretation with respect to PA, but not in the one with respect to ZF. ....
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Undecidability in diagonalizable algebras, Journal of Symbolic Logic, 62, pp. 79--116. C. Smory' nski
The Logic of Provability - Japaridze, de Jongh (1998) (13 citations) (Correct)
....of pairs of arithmetic theories that come to mind, but, unfortunately, are far from being an exhaustive list of all bimodal provability logics. The best known system is the logic PRLPA;ZF discovered by Carlson [1986] and independently (with a different interpretation in mind) by Montagna [1987]. This The Logic of Provability 493 logic can be axiomatized over CS by the principle of essential reflexivity 4(2A A) It is the only bimodal provability logic of type S and a maximal one among the bimodal logics for pairs of sound theories. In other words, PRL T;U = PRLPA;ZF , whenever the ....
....the question Is the Rosser sentence unique does have a definite answer. Hence also, uniqueness of fixed points is not provable in R. Finally, they showed that also the existence part of the fixed point theorem fails for R. Simpler proofs for the completeness theorems were given in de Jongh [1987] and Voorbraak [1988] There are connections between this work in provability logic and speed up. First, de Jongh and Montagna [1988,1989] gave a new simpler proof of Parikh [1971] s theorem that, for any provably recursive function g there is a sentence ff provable in PA such that PA proves PrPA ....
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Provability in finite subtheories of PA, Journal of Symbolic Logic, 52, pp. 494--511.
The Logic of Provability - Japaridze, de Jongh (1997) (13 citations) (Correct)
....but, unfortunately, are far from being an exhausitive list of all bimodal provability logics. Draft April 17, 376 Giorgi Japaridze and Dick de Jongh The best known system is the logic PRLPA;ZF discovered by Carlson [1986] and independently (with a different interpretation in mind) by Montagna [1987]. This logic can be axiomatized over CS by the principle of essential reflexivity 4(2A A) It is the only bimodal provability logic of type S and a maximal one among the bimodal logics for pairs of sound theories. In other words, PRL T;U = PRLPA;ZF , whenever the theories T; U are sound and U ....
....the question Is the Rosser sentence unique does have a definite answer. Hence also, uniqueness of fixed points is not provable in R. Finally, they showed that also the existence part of the fixed point theorem fails for R. Simpler proofs for the completeness theorems were given in de Jongh [1987] and Voorbraak [1988] There are connections between this work in provability logic and speed up. First, de Jongh and Montagna [1988,1989] gave a new simpler proof of Parikh [1971] s theorem that, for any provably recursive function g there is a sentence ff provable in PA such that PA proves PrPA ....
[Article contains additional citation context not shown here]
Provability in finite subtheories of PA, Journal of Symbolic Logic, 52, pp. 494--511.
Chapter Vii - The Logic Of (1968) (1 citation) (Correct)
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Undecidability in diagonalizable algebras, Journal of Symbolic Logic, 62, pp. 79--116. C. Smory' nski
Chapter Vii - The Logic Of (1968) (1 citation) (Correct)
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Provability in finite subtheories of PA, Journal of Symbolic Logic, 52, pp. 494--511.
Soundness Cut-Elimination for NaDSyL - Gilmore (1997) (Correct)
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Resolution in type theory, Journal of Symbolic Logic, vol. 36, 414-432. Apostoli, Peter
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