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Applications of cut-free infinitary derivations to generalized recursion theory
1998
Annals of Pure and Applied Logic
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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
1977
Bulletin of the American Mathematical Society
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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]
1995
Studies in Logic and the Foundations of Mathematics
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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]
2019
Mathesis Universalis, Computability and Proof
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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]
2017
arXiv
pre-print
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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
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“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]
2021
arXiv
pre-print
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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 of ∆ of 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]
2020
arXiv
pre-print
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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
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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
2016
Annual Conference for Computer Science Logic
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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
2010
2010 25th Annual IEEE Symposium on Logic in Computer Science
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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
2009
Annals of Pure and Applied Logic
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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
2010
Archive for Mathematical Logic
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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]
2002
Proof and System-Reliability
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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
2011
Annals of Pure and Applied Logic
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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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