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Applications of cut-free infinitary derivations to generalized recursion theory

Arnold Beckmann, Wolfram Pohlers
1998 Annals of Pure and Applied Logic  
In this paper we want to show that also the statical aspect, i.e., the mere existence of cut-free derivations, has consequences which may be viewed to belong to Descriptive Set Theory or Generalized Recursion  ...  Recursion Theory.  ... 
doi:10.1016/s0168-0072(97)00063-8 fatcat:choz6rogzffw3les7kik37bq3e

Book Review: Proof theory

Solomon Feferman
1977 Bulletin of the American Mathematical Society  
T -» A to cut-free derivations ^D* of r -» A.  ...  Thus any cut-free derivation has a direct character.  ... 
doi:10.1090/s0002-9904-1977-14263-8 fatcat:mdbocvmryzbqjezeg5rx2vlb7u

Admissible proof theory and beyond [chapter]

Michael Rathjen
1995 Studies in Logic and the Foundations of Mathematics  
The latter result was achieved by employing the most advanced techniques in this area of research: cut elimination for infinitary calculi of ramified set theory with Π 2 -reflection rules.  ...  Gentzen's constructive use of ordinals as a method of analyzing formal theories has come to be a paradigm for much of proof theory from then on, particularly as exemplified in the work of Schütte, Takeuti  ...  Stationary collapsing is applicable to all of the theories KP + Π n -reflection. To close, we raise the question of how far afield from Π 1 2 comprehension all this is.  ... 
doi:10.1016/s0049-237x(06)80041-8 fatcat:gayxj5nh2zbubhudzxjkci7r2e

On Relating Theories: Proof-Theoretical Reduction [chapter]

Michael Rathjen, Michael Toppel
2019 Mathesis Universalis, Computability and Proof  
In general, the more that such infinitary methods were employed, the farther did proof theory depart from its initial aims and methods, and the closer did it come instead to ongoing developments in recursion  ...  theory, particularly as generalized to admissible sets; in both one makes use of analogues of regular cardinals, as well as "large" cardinals (inaccessible, Mahlo, etc.). ([19]).  ...  was supported by a University Research Scholarship of the University of Leeds.  ... 
doi:10.1007/978-3-030-20447-1_16 fatcat:loptt6g2jfbmtdczu2qypbk2vu

On sequents of Σ formulas [article]

Andre Kornell
2017 arXiv   pre-print
Finally, we show the equivalence of two completeness principles appropriate to a potentialist conception of the universe of sets.  ...  We also show that a natural extension of such a theory proves the validity of intuitionistic reasoning for that theory.  ...  cut-free derivation.  ... 
arXiv:1704.08155v4 fatcat:fvs4uylftzczxe6s3isdkx5pdy

Page 1150 of Mathematical Reviews Vol. , Issue 87c [page]

1987 Mathematical Reviews  
To obtain our consistency result we interpret S in a fragment of infinitary logic (Section 4) and apply a cut elimination argument adapted from W.  ...  For any formula ~, ~“‘@) is the rela- tivization of y to L(a) obtained by restricting all free and bound variables of y to L(a).)  ... 

Reflection ranks via infinitary derivations [article]

Fedor Pakhomov, James Walsh
2021 arXiv   pre-print
In this note we provide a new proof that instead makes use of traditional proof-theoretic techniques, namely, cut-elimination for infinitary derivations.  ...  This provides an alternative characterization of the notion of "proof-theoretic ordinal," which is among the central concepts in proof theory.  ...  ., that P is a proof ofof height α and cut-rank ρ. We write $ α ρ ∆ to mean that there is a P such that P :$ α ρ ∆. We will need one lemma concerning infinitary derivations.  ... 
arXiv:2107.03521v1 fatcat:534dofs6d5exxhogk6rnnva5km

Cut elimination for systems of transparent truth with restricted initial sequents [article]

Carlo Nicolai
2020 arXiv   pre-print
Finally, we establish a direct link between cut-free derivability in infinitary formulations of the systems considered and fixed-point semantics.  ...  of truth rules to formulas in derivations.  ...  3), and culminates with the eliminability of cut in LGT essentially achieved 2 [Zar11] has also presented a cut-elimination argument for an (infinitary) transparent theory of truth over a contraction-free  ... 
arXiv:2006.07940v2 fatcat:tjxqn5acmnghbexdsfcwdzw424

Page 2593 of Mathematical Reviews Vol. , Issue 96e [page]

1996 Mathematical Reviews  
As an application, the cut elimination theorem provides a method to decide whether two deductions are equal.  ...  An #-controlled derivation with respect to an Q € # is one of RS with the assignment of ordinals “controlled” by #. Among the rules of inference are 2-reflection with respect to Q and Cut.  ... 

Infinitary Proof Theory: the Multiplicative Additive Case

David Baelde, Amina Doumane, Alexis Saurin, Marc Herbstritt
2016 Annual Conference for Computer Science Logic  
for lattice logic (purely additive linear logic with least and greatest fixed points) and showed that certain cut reductions converge to a limit cut-free derivation.  ...  One would hope that extending this to infinitary calculi would lead, e.g., to a better understanding of recursion and corecursion in programming languages.  ...  We then turn to cut elimination in Section 4 and show that (fair) cut reductions converge to an infinitary cut free derivation.  ... 
doi:10.4230/lipics.csl.2016.42 dblp:conf/csl/BaeldeDS16 fatcat:vpem436twbaarpzosgwkng6kfu

Infinitary Completeness in Ludics

Michele Basaldella, Kazushige Terui
2010 2010 25th Annual IEEE Symposium on Logic in Computer Science  
In order to extend the completeness theorem of the original ludics to the infinitary setting, we modify the notion of orthogonality by defining it via safety rather than termination of the interaction.  ...  Our purpose is to provide an interactive form of completeness between infinite proofs and infinite models over formulas of infinite depth (that include recursive types), where proofs and models are homogenous  ...  In domain theory, domains for recursive types are obtained by simply solving domain equations.  ... 
doi:10.1109/lics.2010.47 dblp:conf/lics/BasaldellaT10 fatcat:vcnjuqapwvbybgv5zo46sf4gki

Ordinal analysis by transformations

Henry Towsner
2009 Annals of Pure and Applied Logic  
We apply this to a system of inductive definitions with at least the strength of a recursively inaccessible ordinal.  ...  The technique of using infinitary rules in an ordinal analysis has been one of the most productive developments in ordinal analysis.  ...  But the cut-free proofs of primitive recursive formulas are also proofs in IS, so there is a cut-free proof of 0 = 1 in µ 2 .  ... 
doi:10.1016/j.apal.2008.09.011 fatcat:4psauotquvag5ofq74jpazfcwe

An ordinal analysis for theories of self-referential truth

Graham Emil Leigh, Michael Rathjen
2010 Archive for Mathematical Logic  
We will establish the proof-theoretic strength of all the remaining seven theories and relate their arithmetic part to wellknown theories ranging from PA to the theory of Σ 1 1 dependent choice.  ...  The first attempt at a systematic approach to axiomatic theories of truth was undertaken by Friedman and Sheard [11] .  ...  in general, able to remove applications of (T-Cons).  ... 
doi:10.1007/s00153-009-0170-2 fatcat:oo3s4oq4abhlparssdgmxxetci

Proof Theoretic Complexity [chapter]

G. E. Ostrin, S. S. Wainer
2002 Proof and System-Reliability  
More recently, through the work of Bellantoni and Cook (1992) and Leivant (1994) on their different forms of "tiered recursion", there has been a growing interest in developing theories analogous to classical  ...  A weak formal theory of arithmetic is developed, entirely analogous to classical arithmetic but with two separate kinds of variables: induction variables and quantifier variables.  ...  Introduction The classical methods of proof theory (cut elimination or normalization) enable one to read off, from a proof that a recursively defined function terminates everywhere, a bound on its computational  ... 
doi:10.1007/978-94-010-0413-8_12 fatcat:mo6ha6scmzckrljxoqfuqidsae

Quick cut-elimination for strictly positive cuts

Toshiyasu Arai
2011 Annals of Pure and Applied Logic  
The proof is inspired by the quick cut-elimination due to G. Mints. This technique is also applied to fragments of Heyting arithmetic.  ...  In this paper we show that the intuitionistic theory  ID i <ω (SP) for finitely many iterations of strictly positive operators is a conservative extension of Heyting arithmetic.  ...  Also I would like to thank the referee for his or her careful reading and helpful comments.  ... 
doi:10.1016/j.apal.2011.03.002 fatcat:aqc6inwmkfhrphgemz4bkbghtq
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