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Mathematical Logic: Proof Theory, Constructive Mathematics

Samuel Buss, Ulrich Kohlenbach, Michael Rathjen
2014 Oberwolfach Reports  
The workshop "Mathematical Logic: 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.  ... 
doi:10.4171/owr/2014/52 fatcat:u5y4ahfvk5bghnlcezwbyesc4a

Mathematical Logic: Proof Theory, Constructive Mathematics

Samuel Buss, Ulrich Kohlenbach, Michael Rathjen
2011 Oberwolfach Reports  
The workshop "Mathematical Logic: 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.  ... 
doi:10.4171/owr/2011/52 fatcat:ds55c4o3bjdr7eftdpmynpxvnq

Forcing in Proof Theory

Jeremy Avigad
2004 Bulletin of Symbolic Logic  
The latter include ways 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 theoriesIntuitionistic  ... 
doi:10.2178/bsl/1102022660 fatcat:7sr63xqpabcqjdsuazxbi3rude

An Independence Result for Intuitionistic Bounded Arithmetic

Morteza Moniri
2006 Journal of Logic and Computation  
It is shown that the 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.  ... 
doi:10.1093/logcom/exi085 fatcat:gxdlsvxk6fgjbnegws5ua7ap74

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  ... 

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.  ... 

Harrington's conservation theorem redone

Fernando Ferreira, Gilda Ferreira
2008 Archive for Mathematical Logic  
Leo Harrington showed that the second-order 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.  ... 
doi:10.1007/s00153-008-0080-8 fatcat:j5ob4zf3bzghpibbyj42u5zoiu

Proof Theory [article]

Jeremy Avigad
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.  ... 
arXiv:1711.01994v2 fatcat:3zjao7da6vdljgkb66w2vdjl74

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

M. Moniri
2003 Journal of Logic and Computation  
In this note we show that the 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.  ... 
doi:10.1093/logcom/13.6.881 fatcat:six76ujd45ajhprhoq7t4trxfm

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

Tristan Crolard
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.  ... 
doi:10.1016/s0304-3975(99)00124-3 fatcat:kxguirzxanaatl5y2c2ojppsty

Interpreting classical theories in constructive ones

Jeremy Avigad
2000 Journal of Symbolic Logic (JSL)  
A number 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.  ... 
doi:10.2307/2695075 fatcat:2waagpylczbhpmyohpxfqytk7i

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.  ... 

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).  ... 

Saturated models of universal theories

Jeremy Avigad
2002 Annals of Pure and Applied Logic  
A notion called Herbrand saturation is shown to provide the modeltheoretic analogue 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.  ... 
doi:10.1016/s0168-0072(02)00030-1 fatcat:lhuy44jhyrbwrloyawoediuumm
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