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Interpretability logic
 Mathematical Logic, Proceedings of the 1988 Heyting Conference
, 1990
"... Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength ..."
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Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength of theories, or better to prove
Undecidability in diagonalizable algebras
 The Journal of Symbolic Logic
"... For a formal theory T, the diagonalizable algebra (a.k.a. Magari algebra) of T, denoted DT, is the Lindenbaum sentence algebra of T endowed with the unary operator T arising from the provability predicate of T: (the equivalence class of) a sentence ' is sent by T to (the equivalence class of) t ..."
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Cited by 2 (0 self)
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For a formal theory T, the diagonalizable algebra (a.k.a. Magari algebra) of T, denoted DT, is the Lindenbaum sentence algebra of T endowed with the unary operator T arising from the provability predicate of T: (the equivalence class of) a sentence ' is sent by T to (the equivalence class of) the Tsentence expressing that T proves '. It was shown in Shavrukov [6] that the diagonalizable algebras of PA and ZF, as well as the diagonalizable algebras of similarly related pairs of 1sound theories, are not isomorphic. Neither are these algebras rstorder equivalent (Shavrukov [7, Theorem 2.11]). In the present paper we establish a su cient condition, which we name B ( ) coherence, for the diagonalizable algebras of two theories to be isomorphic. It is then immediately seen that DZF = DGB, which answers a question of Smorynski [11]. We also construct nonidentity automorphisms of diagonalizable algebras of all theories under consideration. The techniques we useareacombination of those developed in the
Parameter free induction and reflection
, 1996
"... We give a precise characterization of parameter free n and n induction schemata, I; n and I; n, in terms of reflection principles. This allows us to show that I; n+1 is conservative over I; n w.r.t. boolean combinations of n+1 sentences, for n 1. In particular, we give a positive answer to a questio ..."
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We give a precise characterization of parameter free n and n induction schemata, I; n and I; n, in terms of reflection principles. This allows us to show that I; n+1 is conservative over I; n w.r.t. boolean combinations of n+1 sentences, for n 1. In particular, we give a positive answer to a question by R. Kaye, whether the provably recursive functions of I; 2 are exactly the primitive recursive ones. We also obtain sharp results on the strength of bounded number of instances of parameter free induction in terms of iterated reflection.