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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 13 (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)
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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Cited by 12 (5 self)
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
Comparing hierarchies of total functionals
 Logical Methods in Computer Science, Volume 1, Issue 2, Paper 4 (2005). RICH HIERARCHY 21
"... In this paper, we will address a problem raised by Bauer, Escardó and Simpson. We define two hierarchies of total, continuous functionals over the reals based on domain theory, one based on an “extensional ” representation of the reals and the other on an “intensional ” representation. The problem i ..."
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Cited by 5 (3 self)
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In this paper, we will address a problem raised by Bauer, Escardó and Simpson. We define two hierarchies of total, continuous functionals over the reals based on domain theory, one based on an “extensional ” representation of the reals and the other on an “intensional ” representation. The problem is if these two hierarchies coincide. We will show that this coincidence problem is equivalent to the statement that the topology on the KleeneKreisel continuous functionals of a fixed type induced by all continuous functions into the reals is zerodimensional for each type. As a tool of independent interest, we will construct topological embeddings of the KleeneKreisel functionals into both the extensional and the intensional hierarchy at each type. The embeddings will be hierarchy embeddings as well in the sense that they are the inclusion maps at type 0 and respect application at higher types. 1
Infinite sets that satisfy the principle of omniscience in all varieties of constructive mathematics, MartinLöf formalization, in Agda notation, of part of the paper with the same title
 University of Birmingham, UK, http://www.cs.bham.ac.uk/~mhe/papers/ omniscient/AnInfiniteOmniscientSet.html, September 2011. SETS IN CONSTRUCTIVE MATHEMATICS 21
"... Abstract. We show that there are plenty of infinite sets that satisfy the omniscience principle, in a minimalistic setting for constructive mathematics that is compatible with classical mathematics. A first example of an omniscient set is the onepoint compactification of the natural numbers, also k ..."
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Abstract. We show that there are plenty of infinite sets that satisfy the omniscience principle, in a minimalistic setting for constructive mathematics that is compatible with classical mathematics. A first example of an omniscient set is the onepoint compactification of the natural numbers, also known as the generic convergent sequence. We relate this to Grilliot’s and Ishihara’s Tricks. We generalize this example to many infinite subsets of the Cantor space. These subsets turn out to be ordinals in a constructive sense, with respect to the lexicographic order, satisfying both a wellfoundedness condition with respect to decidable subsets, and transfinite induction restricted to decidable predicates. The use of simple types allows us to reach any ordinal below ɛ0, and richer type systems allow us to get higher. §1. Introduction. We show that there are plenty of infinite sets X that satisfy the omniscience principle for every function p: X → 2, ∃x ∈ X(p(x) = 0) ∨ ∀x ∈ X(p(x) = 1). For X finite this is trivial, and for X = N, this is LPO, the limited principle of omniscience, which of course is and will remain a taboo in any variety of
Reverse mathematics, countable and uncountable: a computational approach
, 2009
"... Reverse mathematics analyzes the complexity of mathematical statements in terms of the strength of axiomatic systems needed to prove them. Its setting is countable mathematics and subsystems of second order arithmetic. We present a similar analysis based on (recursion theoretic) computational comple ..."
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Cited by 1 (0 self)
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Reverse mathematics analyzes the complexity of mathematical statements in terms of the strength of axiomatic systems needed to prove them. Its setting is countable mathematics and subsystems of second order arithmetic. We present a similar analysis based on (recursion theoretic) computational complexity instead. In the countable case, this view is implicit in many of results in the area. By making it explicit and precise, we provide an alternate approach to this type of analysis for countable mathematics. It may be more intelligible to some mathematicians in that it replaces logic and proof systems with relative computability. In the uncountable case, second order arithmetic and its proof theory is insufficient for the desired analysis. Our computational approach, however, supplies a ready made paradigm for similar analyses. It can be implemented with any appropriate notion of computation on uncountable sets.
This document in subdirectoryRS/99/11/
, 1999
"... Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS ..."
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Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS