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Applications of cut-free infinitary derivations to generalized recursion theory

1998
*
Annals of Pure and Applied Logic
*

In this paper we want

doi:10.1016/s0168-0072(97)00063-8
fatcat:choz6rogzffw3les7kik37bq3e
*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*. ...##
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Book Review: Proof theory

1977
*
Bulletin of the American Mathematical Society
*

T -» A

doi:10.1090/s0002-9904-1977-14263-8
fatcat:mdbocvmryzbqjezeg5rx2vlb7u
*to**cut*-*free**derivations*^D**of*r -» A. ... Thus any*cut*-*free**derivation*has a direct character. ...##
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Admissible proof theory and beyond
[chapter]

1995
*
Studies in Logic and the Foundations of Mathematics
*

The latter result was achieved by employing the most advanced techniques in this area

doi:10.1016/s0049-237x(06)80041-8
fatcat:gayxj5nh2zbubhudzxjkci7r2e
*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. ...##
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On Relating Theories: Proof-Theoretical Reduction
[chapter]

2019
*
Mathesis Universalis, Computability and Proof
*

In

doi:10.1007/978-3-030-20447-1_16
fatcat:loptt6g2jfbmtdczu2qypbk2vu
*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. ...##
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On sequents of Σ formulas
[article]

2017
*
arXiv
*
pre-print

Finally, we show the equivalence

arXiv:1704.08155v4
fatcat:fvs4uylftzczxe6s3isdkx5pdy
*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*. ...##
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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).) ...##
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Reflection ranks via infinitary derivations
[article]

2021
*
arXiv
*
pre-print

In this note we provide a new proof that instead makes use

arXiv:2107.03521v1
fatcat:534dofs6d5exxhogk6rnnva5km
*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*. ...##
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Cut elimination for systems of transparent truth with restricted initial sequents
[article]

2020
*
arXiv
*
pre-print

Finally, we establish a direct link between

arXiv:2006.07940v2
fatcat:tjxqn5acmnghbexdsfcwdzw424
*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*...##
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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*. ...##
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Infinitary Proof Theory: the Multiplicative Additive Case

2016
*
Annual Conference for Computer Science Logic
*

for lattice logic (purely additive linear logic with least and greatest fixed points) and showed that certain

doi:10.4230/lipics.csl.2016.42
dblp:conf/csl/BaeldeDS16
fatcat:vpem436twbaarpzosgwkng6kfu
*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*. ...##
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Infinitary Completeness in Ludics

2010
*
2010 25th Annual IEEE Symposium on Logic in Computer Science
*

In order

doi:10.1109/lics.2010.47
dblp:conf/lics/BasaldellaT10
fatcat:vcnjuqapwvbybgv5zo46sf4gki
*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. ...##
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Ordinal analysis by transformations

2009
*
Annals of Pure and Applied Logic
*

We apply this

doi:10.1016/j.apal.2008.09.011
fatcat:4psauotquvag5ofq74jpazfcwe
*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 . ...##
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An ordinal analysis for theories of self-referential truth

2010
*
Archive for Mathematical Logic
*

We will establish the proof-theoretic strength

doi:10.1007/s00153-009-0170-2
fatcat:oo3s4oq4abhlparssdgmxxetci
*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). ...##
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Proof Theoretic Complexity
[chapter]

2002
*
Proof and System-Reliability
*

More recently, through the work

doi:10.1007/978-94-010-0413-8_12
fatcat:mo6ha6scmzckrljxoqfuqidsae
*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 ...##
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Quick cut-elimination for strictly positive cuts

2011
*
Annals of Pure and Applied Logic
*

The proof is inspired by the quick

doi:10.1016/j.apal.2011.03.002
fatcat:aqc6inwmkfhrphgemz4bkbghtq
*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. ...
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