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P. Hájek, P. Pudlák (1993)
Metamathematics of First-Order Arithmetic
J. Shoenfield (1967)
Mathematical logic
P. Clote, P. Hájek, J. Paris (1990)
On some formalized conservation results in arithmeticArchive for Mathematical Logic, 30
W. Sieg (1985)
Fragments of arithmeticAnn. Pure Appl. Log., 28
C. Parsons (1970)
On a Number Theoretic Choice Schema and its Relation to InductionStudies in logic and the foundations of mathematics, 60
R. Kaye, J. Paris, C. Dimitracopoulos (1988)
On parameter free induction schemasJournal of Symbolic Logic, 53
C. Parsons (1968)
Hierarchies of Primitive Recursive FunctionsMathematical Logic Quarterly, 14
J. Paris (1980)
A Hierarchy of Cuts in Models of Arithmetic
H. Ono (1987)
Reflection Principles in Fragments of Peano ArithmeticMath. Log. Q., 33
L. Beklemishev (1995)
Notes on local reflection principlesMolecular & Cellular Proteomics
A. Visser (1991)
The unprovability of small inconsistencyArchive for Mathematical Logic, 32
S. Feferman (1959)
Arithmetization of metamathematics in a general setting
G. Kreisel, A. Levy (1968)
Reflection Principles and Their Use for Establishing the Complexity of Axiomatic SystemsMathematical Logic Quarterly, 14
L. Beklemishev (1995)
Induction Rules, Reflection Principles, and Provably Recursive FunctionsAnn. Pure Appl. Log., 85
A. Visser (1991)
The formalization of InterpretabilityStudia Logica, 50
S. Cook (1990)
Computational complexity of higher type functions
D. Leivant (1983)
The optimality of induction as an axiomatization of arithmeticJournal of Symbolic Logic, 48
C. Parsons (1972)
On n-quantifier inductionJournal of Symbolic Logic, 37
By a result of Paris and Friedman, the collection axiom schema for $\Sigma_{n+1}$ formulas, $B\Sigma_{n+1}$ , is $\Pi_{n+2}$ conservative over $I\Sigma_n$ . We give a new proof-theoretic proof of this theorem, which is based on a reduction of $B\Sigma_n$ to a version of collection rule and a subsequent analysis of this rule via Herbrand's theorem. A generalization of this method allows us to improve known results on reflection principles for $B\Sigma_n$ and to answer some technical questions left open by Sieg [23] and Hájek [9]. We also give a new proof of independence of $B\Sigma_{n+1}$ over $I\Sigma_n$ by a direct recursion-theoretic argument and answer an open problem formulated by Gaifman and Dimitracopoulos [8].
Archive for Mathematical Logic – Springer Journals
Published: Jul 1, 1998
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