Non‐elementary speed‐ups in logic calculi

Arai, Toshiyasu

doi:10.1002/malq.200710067

Cited by 4 publications

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“…[3]. When T has an axiom schema, e.g., T = PA, the Peano arithmetic with the complete induction schema, let us define the fixed point extension FiX(PA) to have the induction schema for any formula with the fixed point predicate Q.…”

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“…By the completeness theorem, it is obvious that the resulting extension of T is conservative over T , though it has a non-elementary speed-up over T when T is a recursive theory containing the elementary recursive arithmetic EA, cf. [3]. When T has an axiom schema, e.g., T = PA, the Peano arithmetic with the complete induction schema, let us define the fixed point extension FiX(PA) to have the induction schema for any formula with the fixed point predicate Q.…”

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Intuitionistic fixed point theories over set theories

Arai

2015

Arch. Math. Logic

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In this paper we show that the intuitionistic fixed point theory FiX i (T ) over set theories T is a conservative extension of T if T can manipulate finite sequences and has the full foundation schema.1 Intuitionistic fixed point theory over set theories T For a theory T in a laguage L, let Q(X, x) be an X-positive formula in the language L ∪ {X} with an extra unary predicate symbol X. Introduce a fresh unary predicate symbol Q together with the axiom stating that Q is a fixed point of Q(X, x):By the completeness theorem, it is obvious that the resulting extension of T is conservative over T , though it has a non-elementary speed-up over T when T is a recursive theory containing the elementary recursive arithmetic EA, cf.[3]. When T has an axiom schema, e.g., T = PA, the Peano arithmetic with the complete induction schema, let us define the fixed point extension FiX(PA) to have the induction schema for any formula with the fixed point predicate Q. Then FiX(PA) is stronger than PA, e.g., FiX(PA) proves the consistency of PA. For the proof-theretic strength of the fixed point theory FiX(PA), see [2,10,16].On the other side, W. Buchholz [15] shows that an intuitionistic fixed point theory over the intuitionistic (Heyting) arithmetic HA for strongly positive formulae Q(X, x) is proof-theoretically reducible to HA. In a language of arithmetic strongly positive formulae with respect to X are generated from arithmetic formulae and atomic ones X(t) by means of positive connectives ∨, ∧, ∃, ∀. Then Rüede and Strahm [18] extends the result to the intuitionistic fixed point theory FiX i (HA) for strictly positive formulae Q(X, x), in which the predicate symbol X does not occur in the antecedent ϕ of implications ϕ → ψ nor in the scope 1

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confidence: 99%