Results 1  10
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13
Strongly Uniform Bounds from SemiConstructive Proofs
, 2004
"... In [12], the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie nonconstructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and nor ..."
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Cited by 10 (7 self)
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In [12], the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie nonconstructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and normed linear spaces and guarantee the independence of the bounds from parameters raging over metrically bounded (not necessarily compact!) spaces. The use of classical logic imposes some severe restrictions on the formulas and proofs for which the extraction can be carried out. In this paper we consider similar metatheorems for semiintuitionistic proofs, i.e. proofs in an intuitionistic setting enriched with certain nonconstructive principles. Contrary to
2001), Intuitionistic choice and restricted classical logic
 Mathematical Logic Quarterly
"... König’s lemma, primitive recursive arithmetic. ..."
On weak Markov's principle
 MLQ MATH. LOG. Q
, 2002
"... We show that the socalled weak Markov's principle (WMP) which states that every pseudopositive real number is positive is underivable in T # :=EHA # +AC. Since T # allows to formalize (at least large parts of) Bishop's constructive mathematics this makes it unlikely that WMP can ..."
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Cited by 4 (1 self)
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We show that the socalled weak Markov's principle (WMP) which states that every pseudopositive real number is positive is underivable in T # :=EHA # +AC. Since T # allows to formalize (at least large parts of) Bishop's constructive mathematics this makes it unlikely that WMP can be proved within the framework of Bishopstyle mathematics (which has been open for about 20 years). The underivability even holds if the ine#ective schema of full comprehension (in all types) for negated formulas (in particular for #free formulas) is added which allows to derive the law of excluded middle for such formulas.
PROOF INTERPRETATIONS AND MAJORIZABILITY
"... Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpret ..."
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Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpretation of Ulrich Kohlenbach, has been at the center of a vigorous program in applied proof theory dubbed proof mining. We discuss some of the traditional and majorizability interpretations, including the recent bounded interpretations, and focus on the main theoretical techniques behind proof mining. Contents
S.: NonCommutative Infinitary Peano Arithmetic
 In: Proceedings of CSL 2011
, 2011
"... Does there exist any sequent calculus such that it is a subclassical logic and it becomes classical logic when the exchange rules are added? The first contribution of this paper is answering this question for infinitary Peano arithmetic. This paper defines infinitary Peano arithmetic with noncommut ..."
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Does there exist any sequent calculus such that it is a subclassical logic and it becomes classical logic when the exchange rules are added? The first contribution of this paper is answering this question for infinitary Peano arithmetic. This paper defines infinitary Peano arithmetic with noncommutative sequents, called noncommutative infinitary Peano arithmetic, so that the system becomes equivalent to Peano arithmetic with the omegarule if the the exchange rule is added to this system. This system is unique among other noncommutative systems, since all the logical connectives have standard meaning and specifically the commutativity for conjunction and disjunction is derivable. This paper shows that the provability in noncommutative infinitary Peano arithmetic is equivalent to Heyting arithmetic with the recursive omega rule and the law of excluded middle for Sigma01 formulas. Thus, noncommutative infinitary Peano arithmetic is shown to be a subclassical logic. The cut elimination theorem in this system is also proved. The second contribution of this paper is introducing infinitary Peano arithmetic having antecedentgrouping and no right exchange rules. The first contribution of this paper is achieved through this system. This system is obtained from the positive fragment of infinitary Peano arithmetic without the exchange rules by extending it from a positive fragment to a full system, preserving its 1backtracking game semantics. This paper shows that this system is equivalent to both noncommutative infinitary Peano arithmetic, and Heyting arithmetic with the recursive omega rule and the Sigma01 excluded middle.
Bounded Modified Realizability
, 2005
"... We define a notion of realizability, based on a new assignment of formulas, which does not care for precise witnesses of existential statements, but only for bounds for them. The novel form of realizability supports a very general form of the FAN theorem, refutes Markov’s principle but meshes well w ..."
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Cited by 1 (1 self)
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We define a notion of realizability, based on a new assignment of formulas, which does not care for precise witnesses of existential statements, but only for bounds for them. The novel form of realizability supports a very general form of the FAN theorem, refutes Markov’s principle but meshes well with some classical principles, including the lesser limited principle of omniscience and weak König’s lemma. We discuss some applications, as well as some previous results in the literature. 1
unknown title
"... and/or the philosophy of mathematics? At the age of 13 or so some initial interest in philosophy and Aristotelian logic was prompted by my classes in Ancient Greek language which was a main emphasis of study at my high school. My real interest in the foundations of mathematics, however, started at t ..."
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and/or the philosophy of mathematics? At the age of 13 or so some initial interest in philosophy and Aristotelian logic was prompted by my classes in Ancient Greek language which was a main emphasis of study at my high school. My real interest in the foundations of mathematics, however, started at the age of 17 during my last year at high school. Our mathematics teacher had the idea to have each of us to write an extended essay on some period in the history of mathematics. He designed a list of 20 topics starting from ancient mathematics to the beginning 20th century. The very day the topics could be chosen I was ill and could not attend school. When I finally was back in school I had to learn that only topic no. 20 on ‘Cantor, Dedekind, Hilbert ’ was left, apparently because everybody had figured out that a topic touching on comparatively recent mathematics would be more difficult to deal with than, say, Babylonian mathematics. After I had overcome some initial shock I went to the university library in Frankfurt to get hold of the collected works of G. Cantor, D. Hilbert as well as R. Dedekind’s ‘Was sind und was sollen die Zahlen ’ and some popular treatments of the ‘foundational crisis’ at the early 20’s century. Immediately, I got excited about the topic. After having finished the essay I was determined to study philosophy and mathematics with the aim to become a logician.
Bounded Functional Interpretation
"... We present a new functional interpretation, based on a novel assignment of formulas. In contrast with Gödel’s functional “Dialectica ” interpretation, the new interpretation does not care for precise witnesses of existential statements, but only for bounds for them. New principles are supported by o ..."
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We present a new functional interpretation, based on a novel assignment of formulas. In contrast with Gödel’s functional “Dialectica ” interpretation, the new interpretation does not care for precise witnesses of existential statements, but only for bounds for them. New principles are supported by our interpretation, including (a version of) the FAN theorem, weak König’s lemma and the lesser limited principle of omniscience. Conspicuous among these principles are also refutations of some laws of classical logic. Notwithstanding, we end up discussing some applications of the new interpretation to theories of classical arithmetic and analysis.