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Mathematical Logic: Proof Theory, Constructive Mathematics

2014
*
Oberwolfach Reports
*

The workshop "Mathematical Logic:

doi:10.4171/owr/2014/52
fatcat:u5y4ahfvk5bghnlcezwbyesc4a
*Proof**Theory*,*Constructive*Mathematics" was centered around*proof*-*theoretic*aspects*of*current mathematics,*constructive*mathematics and logical aspects*of*computational ... Beklemishev (joint with Evgeny Dashkov) Lev Gordeev*Proof*-*theoretic**conservations**of**weak**weak**intuitionistic**constructive**set**theories*. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... Other talks dealt with new*conservation*results for systems*of**constructive**set**theory*(L. ...##
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Mathematical Logic: Proof Theory, Constructive Mathematics

2011
*
Oberwolfach Reports
*

The workshop "Mathematical Logic:

doi:10.4171/owr/2011/52
fatcat:ds55c4o3bjdr7eftdpmynpxvnq
*Proof**Theory*,*Constructive*Mathematics" was centered around*proof*-*theoretic*aspects*of*current mathematics,*constructive*mathematics and logical aspects*of*computational ... Beklemishev (joint with Evgeny Dashkov) Lev Gordeev*Proof*-*theoretic**conservations**of**weak**weak**intuitionistic**constructive**set**theories*. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... Other talks dealt with new*conservation*results for systems*of**constructive**set**theory*(L. ...##
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Forcing in Proof Theory

2004
*
Bulletin of Symbolic Logic
*

The latter include ways

doi:10.2178/bsl/1102022660
fatcat:7sr63xqpabcqjdsuazxbi3rude
*of*obtaining*conservation*results for classical and*intuitionistic**theories*, interpreting classical*theories*in*constructive*ones, and constructivizing model-*theoretic*arguments ... AbstractPaul Cohen's method*of*forcing, together with Saul Kripke's related semantics for modal and*intuitionistic*logic, has had profound effects on a number*of*branches*of*mathematical logic, from*set*... ) Hilbert-style*proof**theory*: • Formalize mathematical reasoning • Understand infinitary reasoning in explicit,*constructive*terms In contrast to forcing in*set**theory*: • Weaker*theories*•*Intuitionistic*...##
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An Independence Result for Intuitionistic Bounded Arithmetic

2006
*
Journal of Logic and Computation
*

It is shown that the

doi:10.1093/logcom/exi085
fatcat:gxdlsvxk6fgjbnegws5ua7ap74
*intuitionistic**theory**of*polynomial induction on positive Π b 1 (coNP) formulas does not prove the sentence ¬¬∀x, y∃z ≤ y(x ≤ |y| → x = |z|). ... The above independence result is proved by*constructing*an ω-chain*of*submodels*of*a countable model*of*S 2 + Ω 3 + ¬exp such that none*of*the worlds in the chain satisfies the sentence, and interpreting ...*Proof*See [J1, Proposition 8] and [T] . A model*theoretic**construction*and its application In this section we work in the language*of*BASIC. ...##
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Page 7807 of Mathematical Reviews Vol. , Issue 2002K
[page]

2002
*
Mathematical Reviews
*

We also investigate the

*proof*-*theoretic*as well as the computational strength*of*UWKL relative to the*intuitionistic*variant*of*PRA” both with and without the Markov principle.” 2002k:03105 03F35 03F0S ... It is also noted that the*theories*iop, iV; and ill, are closed under Friedman’s translation*Proof**theory*and*constructive*mathematics 2002k:03105 by negated formulas and so under Visser’s rule VR and ...##
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Page 8473 of Mathematical Reviews Vol. , Issue 2004k
[page]

2004
*
Mathematical Reviews
*

This paper shows via a model

*theoretic**proof*that if one considers*intuitionistic**theories*the analogous result does not hold. ... In the paper under review a semantic*proof**of*Léb’s theorem for*theories*T containg ZF*set**theory*is presented. Without using the diagonalization lemma, a sentence AUT, is*constructed*. ...##
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Harrington's conservation theorem redone

2008
*
Archive for Mathematical Logic
*

Leo Harrington showed that the second-order

doi:10.1007/s00153-008-0080-8
fatcat:j5ob4zf3bzghpibbyj42u5zoiu
*theory**of*arithmetic WKL 0 is Π 1 1 -*conservative*over the*theory*RCA 0 . Harrington's*proof*is modeltheoretic, making use*of*a forcing argument. ... A purely*proof*-*theoretic**proof*, avoiding forcing, has been eluding the efforts*of*researchers. In this short paper, we present a*proof**of*Harrington's result using a cut-elimination argument. ... [3] ) and*weak*König's lemma. It is natural to ask whether a Harrington type*conservation*result holds in the*intuitionistic**setting*. ...##
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Proof Theory
[article]

2017
*
arXiv
*
pre-print

*Proof*

*theory*began in the 1920's as a part

*of*Hilbert's program, which aimed to secure the foundations

*of*mathematics by modeling infinitary mathematics with formal axiomatic systems and proving those ... Today such a viewpoint has applications in mathematics, computer science, and the philosophy

*of*mathematics. ... RCA 0 : a

*weak*base

*theory*,

*conservative*over primitive recursive arithmetic, with a recursive comprehension axiom, that is, a principle

*of*comprehension for recursive (computable)

*sets*. 2. ...

##
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Comparing Constructive Arithmetical Theories Based on NP-PIND and coNP-PIND

2003
*
Journal of Logic and Computation
*

In this note we show that the

doi:10.1093/logcom/13.6.881
fatcat:six76ujd45ajhprhoq7t4trxfm
*intuitionistic**theory**of*polynomial induction on Π b+ 1 -formulas does not imply the*intuitionistic**theory*IS 1 2*of*polynomial induction on Σ b+ 1 -formulas. ... Similar results hold also for length induction in place*of*polynomial induction. We also investigate the relation between various other*intuitionistic*first-order*theories**of*bounded arithmetic. ... Acknowledgements Final version*of*this paper to appear in the Journal*of*Logic and Computation. ...##
###
Page 5797 of Mathematical Reviews Vol. , Issue 91K
[page]

1991
*
Mathematical Reviews
*

These

*theories*fall into three families:*constructive**set**theory*(introduced by Myhill and Friedman), theo- ries*of*types (introduced by Martin-Léf), and*theories**of*operations and classes (introduced ... Each family contains theo- ries formulated with only ‘restricted induction’, and these*theories*all turn out to be*conservative*over*intuitionistic*arithmetic HA. ...##
###
Subtractive logic

2001
*
Theoretical Computer Science
*

*of*

*intuitionistic*logic with a new connector (subtraction) dual to implication. ... We examine the propositional calculus underlying the type system

*of*bicartesian closed categories with coexponents and we show that this calculus corresponds to subtractive logic: a

*conservative*extension ... They correspond, respectively, to the

*set*-

*theoretical*notions

*of*singleton and empty

*set*, cartesian product and disjoint union. Then we consider the

*construction*dual to the exponent. 1.2.1. ...

##
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Interpreting classical theories in constructive ones

2000
*
Journal of Symbolic Logic (JSL)
*

A number

doi:10.2307/2695075
fatcat:2waagpylczbhpmyohpxfqytk7i
*of*classical*theories*are interpreted in analogous*theories*that are based on*intuitionistic*logic. ... The classical*theories*considered include subsystems*of*first- and second-order arithmetic, bounded arithmetic, and admissible*set**theory*. ... I should emphasize that the*proof*-*theoretic*equivalences*of*the classical and*intuitionistic**theories*discussed here are well-known. ...##
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Page 7297 of Mathematical Reviews Vol. , Issue 2003j
[page]

2003
*
Mathematical Reviews
*

The authors present many clearly proved results concerning algebraic and probabilistic properties

*of*03F*Proof**theory*and*constructive*mathematics 2003):03077 maximal fuzzy codes. ... In this paper, various*set**theoretic*properties*of*rough-fuzzy functions are exploited to characterize the concept*of*rough-fuzzy*sets*. ...##
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Page 2541 of Mathematical Reviews Vol. , Issue 84g
[page]

1984
*
Mathematical Reviews
*

This article elaborates the

*construction**of*Heyting-valued models for*intuitionistic**set**theory*[cf. R. J. ... The main*proof*-*theoretic*result*of*this paper is: =!,.,-DC+BR is a*conservative*extension*of*Aut(II') for II!-sentences, where i= min(n+ 2, 4). ...##
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Saturated models of universal theories

2002
*
Annals of Pure and Applied Logic
*

A notion called Herbrand saturation is shown to provide the modeltheoretic analogue

doi:10.1016/s0168-0072(02)00030-1
fatcat:lhuy44jhyrbwrloyawoediuumm
*of*a*proof*-*theoretic*method, Herbrand analysis, yielding uniform model-*theoretic**proofs**of*a number*of*important*conservation*... A*constructive*, algebraic variation*of*the method is described, providing yet a third approach, which is finitary but retains the semantic flavor*of*the model-*theoretic*version. ... Using cut-elimination or normalization to prove the Π 2 soundness*of*the*weak**constructive**theory*, one then obtains finitary*proofs**of*the*conservation*results. ...
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