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Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show t ..."
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Cited by 17 (5 self)
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Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1
Foundational and mathematical uses of higher types
 REFLECTIONS ON THE FOUNDATIONS OF MATHEMATICS: ESSAY IN HONOR OF SOLOMON FEFERMAN
, 1999
"... In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles ..."
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Cited by 11 (4 self)
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In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles which generalize (and for n = 0 coincide with) the socalled `weak' König's lemma WKL (which has been studied extensively in the context of second order arithmetic) to logically more complex tree predicates. Whereas the second order context used in the program of reverse mathematics requires an encoding of higher analytical concepts like continuous functions F : X ! Y between Polish spaces X;Y , the more exible language of our systems allows to treat such objects directly. This is of relevance as the encoding of F used in reverse mathematics tacitly yields a constructively enriched notion of continuous functions which e.g. for F : IN ! IN can be seen (in our higher order context)
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 be proved ..."
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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.
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.
Classical provability of uniform versions and intuitionistic provability
, 2013
"... Along the line of HirstMummert [9] and Dorais [4], we analyze the relationship between the classical provability of uniform versions Uni(S) of Π2statements S with respect to higher order reverse mathematics and the intuitionistic provability of S. Our main theorem states that (in particular) for e ..."
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Along the line of HirstMummert [9] and Dorais [4], we analyze the relationship between the classical provability of uniform versions Uni(S) of Π2statements S with respect to higher order reverse mathematics and the intuitionistic provability of S. Our main theorem states that (in particular) for every Π2statement S of some syntactical form, if its uniform version derives the uniform variant of ACA over a classical system of arithmetic in all finite types with weak extensionality, then S is not provable in strong semiintuitionistic systems including bar induction BI in all finite types but also nonconstructive principles such as König’s lemma KL and uniform weak König’s lemma UWKL. Our result is applicable to many mathematical principles whose sequential versions imply ACA. 1
Realizability and Strong Normalization for a CurryHoward Interpretation of HA + EM1
"... We present a new CurryHoward correspondence for HA + EM1, constructive Heyting Arithmetic with the excluded middle on Σ 0 1formulas. We add to the lambda calculus an operator ‖a which represents, from the viewpoint of programming, an exception operator with a delimited scope, and from the viewpoin ..."
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We present a new CurryHoward correspondence for HA + EM1, constructive Heyting Arithmetic with the excluded middle on Σ 0 1formulas. We add to the lambda calculus an operator ‖a which represents, from the viewpoint of programming, an exception operator with a delimited scope, and from the viewpoint of logic, a restricted version of the excluded middle. We motivate the restriction of the excluded middle by its use in proof mining; we introduce new techniques to prove strong normalization for HA + EM1 and the witness property for simply existential statements. One may consider our results as an application of the ideas of Interactive realizability, which we have adapted to the new setting and used to prove our main theorems.