Relative interpretations, Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 7, pp. 146--153. R. Parikh  Home/Search   Document Not in Database   Summary   Related Articles   Check

....more detail. I arrived to the problem of finite consistencies through interpretability of theories. The problem was to characterize interpretability of theories containing a fragment of arithmetic using provability of restricted consistencies. This had been done long before for reflexive theories [35, 36, 20]; the main examples of reflexive theories are Peano Arithmetic and Zermello Fraenkel Set Theory. The property of being reflexive narrows the class of theories too much, e.g. any such theory is not finitely axiomatizable. The solution is to take the restricted consistency statements of Paris and ....

S. Orey. Relative interpretations. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 7:146--153, 1961.

....that T is consistent then proves that as soon as is a theorem of T , is true in M , and the clause (a) of the lemma expresses just this fact; as for clause (b) it is an immediate consequence of (a) for PA Compl ff : Pr ff (p q) Pr ff (p: q) and PA t: t . 12.7. Theorem. Orey [1961], H ajek [1971,1972] PA : For superarithmetic theories T and S the following are equivalent: i) T is interpretable in S , ii) for all m, S Con T#m , iii) T is Pi 1 conservative over S . Proof. i) ii) Suppose t = h ; oei is an interpretation of T in S . Let be the conjunction ....

....implies as a corollary a uniform version of the arithmetic completeness of ILM with regard to PA. For a further consequence, let us first note that the existence of Orey sentences in PA, i.e. arithmetic sentences A such that both PA A and PA :A are interpretable in PA (first obtained by Orey [1961]) follows immediately from the arithmetic completeness of ILM with regard to PA. In Strannegard s terminology this can be phrased as: Orey [1961] showed that the set f p; pg is realistic. Orey continued by asking what similar sets (such as f p, q, p q) p) q) ....

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Relative interpretations, Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 7, pp. 146--153. R. Parikh

....that T is consistent then proves that as soon as is a theorem of T , is true in M , and the clause (a) of the lemma expresses just this fact; as for clause (b) it is an immediate consequence of (a) for PA Compl ff : Pr ff (p q) Pr ff (p: q) and PA t: t . 12.7. Theorem. Orey [1961], H ajek [1971,1972] PA : For superarithmetic theories T and S the following are equivalent: i) T is interpretable in S , ii) for all m, S Con T#m , iii) T is Pi 1 conservative over S . Proof. i) ii) Suppose t = h ; oei is an interpretation of T in S . Let be the conjunction ....

Relative interpretations, Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 7, pp. 146--153. R. Parikh

.... A ) 2A ) 2A 2 E ) PA i(A) and 6 A ) 6 2A ) 2A 2 E ) PA 6 i(A) 2 For the last corollary, we need a definition: ffl Let an Orey set be a set E such that for all A, if A 2 E , then A is either of the form B C or of the form : B C) for some formulas of propositional logic B and C. In [Ore61], Orey asks a question that could be phrased thus in our present terminology: What (finite) Orey sets are realizable . We can answer this question as follows: Corollary 4.4 Suppose E is an r. e. Orey set. Then E is realizable iff it is consistent with ILM . Proof. This follows from the ....

S. Orey. Relative interpretations. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 7:146--153, 1961.

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Relative interpretations, Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 7, pp. 146--153. R. Parikh

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