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48
On the Arithmetical Content of Restricted Forms of Comprehension, Choice and General Uniform Boundedness
 PURE AND APPLIED LOGIC
, 1997
"... In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems T n in all finite types which are suited to formalize substantial parts of analysis but ..."
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In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems T n in all finite types which are suited to formalize substantial parts of analysis but nevertheless have provably recursive function(al)s of low growth. We reduce the use of instances of these principles in T n proofs of a large class of formulas to the use of instances of certain arithmetical principles thereby determining faithfully the arithmetical content of the former. This is achieved using the method of elimination of Skolem functions for monotone formulas which was introduced by the author in a previous paper. As
2001), Intuitionistic choice and restricted classical logic
 Mathematical Logic Quarterly
"... König’s lemma, primitive recursive arithmetic. ..."
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Hybrid Functional Interpretations
"... Abstract. We show how different functional interpretations can be combined via a multimodal linear logic. A concrete hybrid of Kreisel’s modified realizability and Gödel’s Dialectica is presented, and several small applications are given. We also discuss how the hybrid interpretation relates to var ..."
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Abstract. We show how different functional interpretations can be combined via a multimodal linear logic. A concrete hybrid of Kreisel’s modified realizability and Gödel’s Dialectica is presented, and several small applications are given. We also discuss how the hybrid interpretation relates to variants of Dialectica and modified realizability with noncomputational quantifiers. 1
A Note on Spector's QuantifierFree Rule of Extensionality
 Arch. Math. Logic
, 1999
"... In this note we show that the socalled weakly extensional arithmetic in all nite types, which is based on a quanti erfree rule of extensionality due to C. Spector and which is of signi cance in the context of Godel's functional interpretation, does not satisfy the deduction theorem fo ..."
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Cited by 6 (3 self)
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In this note we show that the socalled weakly extensional arithmetic in all nite types, which is based on a quanti erfree rule of extensionality due to C. Spector and which is of signi cance in the context of Godel's functional interpretation, does not satisfy the deduction theorem for additional axioms. This holds already for 1  axioms. Previously, only the failure of the stronger deduction theorem for deductions from (possibly open) assumptions (with parameters kept xed) was known.
Synthesis of moduli of uniform continuity by the Monotone Dialectica Interpretation
 in the proofsystem MINLOG. Electronic Notes in Theoretical Computer Science
, 2007
"... We extract on the computer a number of moduli of uniform continuity for the first few elements of a sequence of closed terms t of Gödel’s T of type (N→N)→(N→N). The generic solution may then be quickly inferred by the human. The automated synthesis of such moduli proceeds from a proof of the heredi ..."
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We extract on the computer a number of moduli of uniform continuity for the first few elements of a sequence of closed terms t of Gödel’s T of type (N→N)→(N→N). The generic solution may then be quickly inferred by the human. The automated synthesis of such moduli proceeds from a proof of the hereditarily extensional equality (≈) of t to itself, hence a proof in a weakly extensional variant of BergerBuchholzSchwichtenberg’s system Z of t ≈(N→N)→(N→N) t. We use an implementation on the machine, in Schwichtenberg’s MinLog proofsystem, of a nonliteral adaptation to Natural Deduction of Kohlenbach’s monotone functional interpretation. This new version of the Monotone Dialectica produces terms in NbEnormal form by means of a recurrent partial NbEnormalization. Such partial evaluation is strictly necessary.
On the Computational Content of the BolzanoWeierstraß Principle
, 2009
"... We will apply the methods developed in the field of ‘proof mining’ to the BolzanoWeierstraß theorem BW and calibrate the computational contribution of using this theorem in proofs of combinatorial statements. We provide an explicit solution of the Gödel functional interpretation (combined with nega ..."
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We will apply the methods developed in the field of ‘proof mining’ to the BolzanoWeierstraß theorem BW and calibrate the computational contribution of using this theorem in proofs of combinatorial statements. We provide an explicit solution of the Gödel functional interpretation (combined with negative translation) as well as the monotone functional interpretation of BW for the product space ∏i∈N[−k i, k i] (with the standard product metric). This results in optimal program and bound extraction theorems for proofs based on fixed instances of BW, i.e. for BW applied to fixed sequences in ∏i∈N[−k i, k i].
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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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.
Unifying functional interpretations
 Notre Dame J. Formal Logic
"... Abstract. The purpose of this article is to present a parametrised functional interpretation. Depending on the choice of the parameter relations one obtains wellknown functional interpretations, such as Gödel’s Dialectica interpretation, DillerNahm’s variant of the Dialectica interpretation, Kohle ..."
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Abstract. The purpose of this article is to present a parametrised functional interpretation. Depending on the choice of the parameter relations one obtains wellknown functional interpretations, such as Gödel’s Dialectica interpretation, DillerNahm’s variant of the Dialectica interpretation, Kohlenbach’s monotone interpretations, Kreisel’s modified realizability and Stein’s family of functional interpretations. A functional interpretation consists of a formula translation and a proof translation. We show that all these interpretation only differ on two choices: firstly, on “how much ” of the counterexamples for A became witnesses for ¬A when defining the formula translation, and, secondly, “how much ” of the witnesses of A one is interested in when defining the proof translation.