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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
Peano's Smart Children  A provability logical study of systems with builtin consistency
"... ..."
Notes on local reflection principles
 UNIVERSITY OF UTRECHT
, 1995
"... We study the hierarchy of reflection principles obtained by restricting the full local reflection schema to the classes of the arithmetical hierarchy. Optimal conservation results w.r.t. the arithmetical complexity for such principles are obtained. ..."
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We study the hierarchy of reflection principles obtained by restricting the full local reflection schema to the classes of the arithmetical hierarchy. Optimal conservation results w.r.t. the arithmetical complexity for such principles are obtained.
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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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
Logic Group 'reprint Series
, 1995
"... Abstract. For finitely axiomatized sequential theories F and reflexive theories R, we give a characterization of the relation `F interprets R ' in terms of provability of restricted consistency statements on cuts. This characterization is used in a proof that the set of IIl (as well as E1) sent ..."
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Abstract. For finitely axiomatized sequential theories F and reflexive theories R, we give a characterization of the relation `F interprets R ' in terms of provability of restricted consistency statements on cuts. This characterization is used in a proof that the set of IIl (as well as E1) sentences yr s.t. GB interprets ZF + Tr is E3complete. Relative interpretability among formal theories has been particularly well studied in two specific cases: that of finitely axiomatized sequential theories (see Smorynski [14], Pudlak [11], Visser [16] etc.), and of reflexive, esp. essentially reflexive theories (see Lindstrom [7], [8] etc.). We have nice characterizations of the interpretability relation between a pair of theories