Proof Theory of Weak Compactness

Arai, Toshiyasu

doi:10.1142/s0219061313500037

Cited by 9 publications

(12 citation statements)

References 13 publications

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“…We need Theorem 1.2 in [6][7][8][9] for proof-theoretic analyses of set theories for weakly compact cardinals, first-order reflecting ordinals, ZF and second-order indescribable cardinals. In these analyses, a provability relation H ⊢ α c Γ derived from operator controlled derivations is defined to be a fixed point of a strictly positive formula.…”

Section: Theorem 12 Fix I (T ) Is a Conservative Extension Of Any Sementioning

confidence: 99%

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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Section: Theorem 12 Fix I (T ) Is a Conservative Extension Of Any Sementioning

confidence: 99%

Intuitionistic fixed point theories over set theories

Arai

2015

Arch. Math. Logic

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“…Let us announce an extension of Theorem 2.2 in [6,7] to the indescribable cardinals over ZF + (V = L).…”

Section: Proposition 23 For Any Classmentioning

confidence: 99%

Conservations of First-Order Reflections

Arai¹

2014

J. symb. log.

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“…In [3] we showed that the existence of a weakly compact, i.e., Π 1 1 -indescribable cardinal over the Zermelo-Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations, while in [4] we describe a proof-theoretic bound on definable countable ordinals in ZF. In this paper we do the same reductions for the existence of a Π 1 N -indescribable cardinal, cf.…”

Section: Introductionmentioning

confidence: 99%

Lifting proof theory to the countable ordinals II: second-order indescribable cardinals

Arai

2014

Preprint

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Proof Theory of Weak Compactness

Abstract: We show that the existence of a weakly compact cardinal over the Zermelo-Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations.

Cited by 9 publications

References 13 publications

Intuitionistic fixed point theories over set theories

Intuitionistic fixed point theories over set theories

Conservations of First-Order Reflections

Lifting proof theory to the countable ordinals II: second-order indescribable cardinals

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