The global structure of computably enumerable sets

Cholak, Peter

doi:10.1090/conm/257/04027

Cited by 3 publications

(3 citation statements)

References 23 publications

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“…This work brings descriptive set theory into contact with current developments in various areas of mathematics such as dynamical systems, including ergodic theory and topological dynamics, the theory of topological groups and their representations, operator algebras, abelian and combinatorial group theory, etc. Moreover, it provides new insights in the traditional relationships of descriptive set theory with other areas of mathematical logic, as, for example, with recursion theory, concerning the global structure of Turing degrees (see P. Cholak et al [6]), or with model theory, through the Topological Vaught Conjecture and the general study of the isomorphism relation on countable structures.…”

Section: Descriptive Set Theorymentioning

confidence: 99%

“…At the general level, there are many open problems about the connections between Turing degrees, rates of growth of functions, set theoretic structural properties, definability in arithmetic and the jump classes. For a whole array of specific question in a range of areas, I recommend Cholak et al [6] the proceedings volume of the 1999 AMS Boulder conference on open problems in recursion theory.…”

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confidence: 99%

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The Prospects for Mathematical Logic in the Twenty-First Century

Buss

Kechris²,

Pillay³

et al. 2001

Bull. symb. log.

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Abstract. The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.We can only see a short distance ahead, but we can see plenty there that needs to be done.A. Turing, 1950. §1. Introduction. The annual meeting of the Association for Symbolic Logic held in Urbana-Champaign, June 2000, included a panel discussion on "The Prospects for Mathematical Logic in the Twenty-First Century". The panel discussions included independent presentations by the four panel members, followed by approximately one hour of lively discussion with members of the audience.The main themes of the discussions concerned the directions mathematical logic should or could pursue in the future. Some members of the audience strongly felt that logic needs to find more applications to mathematics; however, there was disagreement as to what kinds of applications were most likely to be possible and important. Many people also felt that applications to computer science will be of great importance. On the other hand, quite a few people, while acknowledging the importance of applications of logic, felt that the most important progress in logic comes from internal developments.It seems safe to presume that the future of mathematical logic will include a multitude of directions and a blend of these various elements. Indeed, it speaks well for the strength of the field that there are multiple compelling directions for future progress. It is to be hoped that logic will be driven both by internal developments and by external applications, Supported in part by NSF grant DMS-9803515.

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Section: Descriptive Set Theorymentioning

confidence: 99%

mentioning

confidence: 99%

The Prospects for Mathematical Logic in the Twenty-First Century

Buss

Kechris²,

Pillay³

et al. 2001

Bull. symb. log.

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“…My recent interest in Issue 1 began in 1999 at a conference in Boulder, Colorado [10]. There I heard a talk by Shmuel Weinberger, a prominent topologist and geometer.…”

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Some Fundamental Issues Concerning Degrees of Unsolvability

Simpson

2008

Computational Prospects of Infinity

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Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT . The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is > 0 and < 0 ′ . The second issue is to find a "smallness property" of an infinite, co-recursively enumerable set A ⊆ ω which ensures that the Turing degree deg T (A) = a ∈ RT is > 0 and < 0 ′ . In order to address these issues, we embed RT into a slightly larger degree structure, Pw, which is much better behaved. Namely, Pw is the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of 2 ω . We define a specific, natural embedding of RT into Pw, and we present some recent and new research results.

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Extension theorems, orbits, and automorphisms of the computably enumerable sets

Cholak

Harrington

2007

Trans. Amer. Math. Soc.

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Abstract. We prove an algebraic extension theorem for the computably enumerable sets, E. Using this extension theorem and other work we then show if A and A are automorphic via Ψ, then they are automorphic via Λ where Λ L * (A) = Ψ and Λ E * (A) is ∆ 0 3 . We give an algebraic description of when an arbitrary set A is in the orbit of a computably enumerable set A. We construct the first example of a definable orbit which is not a ∆ 0 3 orbit. We conclude with some results which restrict the ways one can increase the complexity of orbits. For example, we show that if A is simple and A is in the same orbit as A, then they are in the same ∆ 0 6 -orbit and, furthermore, we provide a classification of when two simple sets are in the same orbit.

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The global structure of computably enumerable sets

Cited by 3 publications

References 23 publications

The Prospects for Mathematical Logic in the Twenty-First Century

The Prospects for Mathematical Logic in the Twenty-First Century

Some Fundamental Issues Concerning Degrees of Unsolvability

Extension theorems, orbits, and automorphisms of the computably enumerable sets

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