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Applications of cut-free infinitary derivations to generalized recursion theory
1998
Annals of Pure and Applied Logic
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Reals which compute little
[chapter]
Logic Colloquium '02
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We investigate combinatorial lowness properties of sets of natural numbers (reals). The real A is super-low if A ≤tt ∅ , and A is jump-traceable if the values of {e} A (e) can be effectively approximated in a sense to be specified. We investigate those properties, in particular showing that super-lowness and jump-traceability coincide within the r.e. sets but none of the properties implies the other within the ω-r.e. sets. Finally we prove that, for any low r.e. set B, there is is a K-trivial set A ≤T B.
doi:10.1017/9781316755723.012
fatcat:wiwlegscvna5dn6e7vqdp755wq
Axioms of generic absoluteness
[chapter]
Logic Colloquium '02
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We give a unified presentation of the set-theoretic axioms of generic absoluteness, we survey the known results regarding their consistency strength and some of their consequences as well as their relationship to other kinds of set-theoretic axioms, and we provide a list of the main open problems.
doi:10.1017/9781316755723.003
fatcat:nena66ajk5awpngrkvbuvi5jhm
Learning and computing in the limit
[chapter]
Logic Colloquium '02
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We explore two analogies between computability theory and a basic model of learning, namely Osherson and Weinsteins model theoretic learning paradigm. First, we build up the theory of model theoretic learning in a way analogous to the way computability theory is built up. We then discuss ∆2-definability of predicates on classes and prove a limit lemma for continuous functionals.
doi:10.1017/9781316755723.017
fatcat:a7gjfkdbdnf5dat45hnea65aji
PCF theory and Woodin cardinals
[chapter]
Logic Colloquium '02
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Pohlers, for their warm hospitality during the Münster meeting. The first author thanks Andreas Liu for his comments on the section on pcf theory of an earlier version of this paper. ...
doi:10.1017/9781316755723.009
fatcat:j7wptqll7fdphlfbyi36wt5hga
Computable versions of the uniform boundedness theorem
[chapter]
Logic Colloquium '02
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We investigate the computable content of the Uniform Boundedness Theorem and of the closely related Banach-Steinhaus Theorem. The Uniform Boundedness Theorem states that a pointwise bounded sequence of bounded linear operators on Banach spaces is also uniformly bounded. But, given the sequence, can we also effectively find the uniform bound? It turns out that the answer depends on how the sequence is "given". If it is just given with respect to the compact open topology (i.e. if just a sequence
doi:10.1017/9781316755723.007
fatcat:ryypcxzexvbi7dypzj63g44xqq
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... of "programs" is given), then we cannot even compute an upper bound of the uniform bound in general. If, however, the pointwise bounds are available as additional input information, then we can effectively compute an upper bound of the uniform bound. Additionally, we prove an effective version of the contraposition of the Uniform Boundedness Theorem: given a sequence of linear bounded operators which is not uniformly bounded, we can effectively find a witness for the fact that the sequence is not pointwise bounded. As an easy application of this theorem we obtain a computable function whose Fourier series does not converge. §1. Introduction. In this paper we want to study the computational content of some theorems of functional analysis. The Uniform Boundedness Theorem is one of the central theorems of functional analysis and it has first been published in Banach's thesis [1]. Theorem 1.1 (Uniform Boundedness Theorem). Let X be a Banach space, Y a normed space and let (T i ) i∈N be a sequence of bounded linear Here, ||T i || := sup ||x||≤1 ||T i x|| denotes the bound of ||T i ||. Roughly speaking, the Uniform Boundedness Theorem states that each pointwise bounded sequence of linear bounded operators is also uniformly bounded. But, given 1991 Mathematics Subject Classification. 03F60, 03D45, 46S30. Key words and phrases. Computable functional analysis, Effective representations. A preliminary extended abstract version of this paper has been published as [7].
Choice principles in constructive and classical set theories
[chapter]
Logic Colloquium '02
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The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.
doi:10.1017/9781316755723.014
fatcat:wnewggrmcjbvdah56jngx5a3ja
"One is a lonely number": logic and communication
[chapter]
Logic Colloquium '02
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Logic is not just about single-agent notions like reasoning, or zero-agent notions like truth, but also about communication between two or more people. What we tell and ask each other can be just as logical as what we infer in Olympic solitude. We show how such interactive phenomena can be studied systematically by merging epistemic and dynamic logic, leading to new types of question. §1. Logic in a social setting.
doi:10.1017/9781316755723.006
fatcat:j3k6up4tqzgffewmn7qps3w4yi
Frontmatter
[chapter]
1993
Mathematische Grundlagen der Informatik
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Wolfram Pohlers Westfälische Wilhelms-Universität Münster R. Oldenbourg Verlag München Wien 1993 Mathematische
Grundlagen
der Informatik
von
Prof. ...
doi:10.1515/9783110703061-fm
fatcat:g5awyivvdjh7rppzcuk63cqr5a
Embedding finite lattices into the computably enumerable degrees — a status survey
[chapter]
Logic Colloquium '02
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We survey the current status of an old open question in classical computability theory: Which finite lattices can be embedded into the degree structure of the computably enumerable degrees? Does the collection of embeddable finite lattices even form a computable set? Two recent papers by the second author show that for a large subclass of the finite lattices, the so-called join-semidistributive lattices (or lattices without so-called "critical triple"), the collection of embeddable lattices
doi:10.1017/9781316755723.010
fatcat:iqljojjyifgqnjtn43w3ftf7qa
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... s a Π 0 2 -set. This paper surveys recent joint work by the authors, concentrating on restricting the number of meets by considering "quasilattices", i.e., finite upper semilattices in which only some meets of incomparable elements are specified. In particular, we note that all finite quasilattices with one meet specified are embeddable; and that the class of embeddable finite quasilattices with two meets specified, while nontrivial, forms a computable set. On the other hand, more sophisticated techniques may be necessary for finite quasilattices with three meets specified. 1991 Mathematics Subject Classification. Primary: 03D25.
BSL volume 6 issue 1 Cover and Front matter
2000
Bulletin of Symbolic Logic
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., 97 In Memoriam: Kurt Schutte, 1909-1998, by WOLFRAM POHLERS 101 F C O N T E N T S
ARTICLES
•
Address at the Princeton University bicentennial conference on problems
of mathematics (December 17 ...
doi:10.1017/s1079898600010088
fatcat:gr2kothle5bjtpqhmkvu64gzay
Preface
2006
Electronical Notes in Theoretical Computer Science
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Pohlers (Institut für mathematische Logik und Grundlagenforschung, Westfälische Wilhelms-Universität, Germany) Vaughan Pratt (Computer Science Department, Stanford University, and Tiqit Computers, USA ...
Mathematics, University of California at Los Angeles, USA) Hiroakira Ono (Japan Advanced Institute of Science and Technology, Japan) Stanley Peters (Department of Linguistics, Stanford University, USA) Wolfram ...
doi:10.1016/j.entcs.2006.07.021
fatcat:tj44tmggijfqjejxrakx2b574e
Page 2260 of Mathematical Reviews Vol. , Issue 85f
[page]
1985
Mathematical Reviews
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Kanovich (Kalinin)
-) V Pn) and -)&Pn)
‘))>(--- (1 Vez) V-- &pn)---)) D (--- ((prdep2)& --
Jager, Gerhard; Pohlers, Wolfram
Eine beweistheoretische Untersuchung von
(A3-CA) + (BI) und verwandter Systeme ...
Page 34 of Mathematical Reviews Vol. , Issue 2000a
[page]
2000
Mathematical Reviews
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Xiaokang Yu (Altoona, PA)
2000a:03106 03F35 Pohlers, Wolfram (D-MUNS-ML; Miinster) Subsystems of set theory and second order number theory. Handbook of proof theory, 209-335, Stud. Logic Found. ...
Page 2299 of Mathematical Reviews Vol. , Issue 99d
[page]
1991
Mathematical Reviews
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Wainer, Hierarchies of provably recursive functions (149-207); Wolfram Pohlers, Subsys- tems of set theory and second order number theory (209-335); Jeremy Avigad and Solomon Feferman, Gédel’s functional ...
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