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Relative to any non-hyperarithmetic set
[article]

2011
*
arXiv
*
pre-print

We prove that there is a structure, indeed a linear ordering, whose degree spectrum is the

arXiv:1110.1907v1
fatcat:v2qnzyoxmnekliihwqjy5kqfni
*set*of all*non*-*hyperarithmetic*degrees. ...*Relative**to**any**non*-*hyperarithmetic*degree In this section we prove Theorem 1.2. 2.1. Discussion. Goncharov, Harizanov, Knight, MacCoy, R. ... Certainly*any*degree computing a copy of M must be*non*-*hyperarithmetic*, and a*non*-*hyperarithmetic*degree can compute every component of M. ...##
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Analytic equivalence relations satisfying hyperarithmetic-is-recursive
[article]

2013
*
arXiv
*
pre-print

We prove, in ZF+Σ^1_2-determinacy, that for

arXiv:1306.1513v2
fatcat:zvqknsr255gjrjcxgnctajvymu
*any*analytic equivalence relation E, the following three statements are equivalent: (1) E does not have perfectly many classes, (2) E satisfies*hyperarithmetic*-is-recursive ... on a cone, and (3)*relative**to*some oracle, for every equivalence class [Y]_E we have that a real X computes a member of the equivalence class if and only if _1^X≥_1^[Y]. ... We need*to*show that*relative**to*every oracle on a cone, there is a*hyperarithmetic*real not E-equivalent*to**any*computable real. ...##
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Finding subsets of positive measure
[article]

2014
*
arXiv
*
pre-print

On the other hand, there are Π^0_2

arXiv:1408.1999v1
fatcat:zi4s6nw2azatvapyheo7nab6uy
*sets*of reals where no*hyperarithmetic*real can define a closed subset of*non*-zero measure. ... We investigate the question how hard it is*to*find such a*set*, in terms of the index*set*complexity, and in terms of the complexity of the parameter needed*to*define such a closed*set*. ... The hierarchies of effective descriptive*set*theory allow for a further ramification of regularity properties.*Any*(boldface) Borel*set*is effectively (lightface) Borel*relative**to*a parameter. ...##
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Computable structures of rank omega_1^CK
[article]

2005
*
arXiv
*
pre-print

We obtain a computable structure of Scott rank omega_1^CK (call this ock), and give a general coding procedure that transforms

arXiv:math/0508507v1
fatcat:bu3gyfj6s5he7enhji37hkknry
*any**hyperarithmetical*structure A into a computable structure A' such that ... A computable (or X-computable) structure A is said*to*be*relatively*intrinsically ∆ 0 α categorical if for*any*copy B of A, there is an isomorphism f from A onto B such that f is ∆ 0 α*relative*B (or X ... Morozov [10] used it*to*produce computable structures which share with the Harrison ordering the feature that there are*non*-trivial automorphisms, but not*hyperarithmetical*ones. ...##
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Equivalence Relations on Classes of Computable Structures
[chapter]

2009
*
Lecture Notes in Computer Science
*

If the index

doi:10.1007/978-3-642-03073-4_21
fatcat:tkc2imruzrc2zg44ygekmrtoe4
*set*of K c is*hyperarithmetical*then (the index*sets*of) such natural equivalence relations as the isomorphism or bi-embeddability relation are Σ 1 1 . ... In the present paper we study the status of these Σ 1 1 equivalence relations (on classes of computable structures with*hyperarithmetical*index*set*) within the class of Σ 1 1 equivalence relations as a ... Let R be the class of all isomorphism relations on classes K c , where K is*any*class of structures with*hyperarithmetical*index*set*. ...##
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Pi01 encodability and omniscient reductions
[article]

2016
*
arXiv
*
pre-print

By a result of Solovay, the computably encodable

arXiv:1603.01086v2
fatcat:brz5d6tobnaqlcy5e62u6hrdyq
*sets*are exactly the*hyperarithmetic*ones. ... In this paper, we extend this notion of computable encodability*to*subsets of the Baire space and we characterize the Π^0_1 encodable compact*sets*as those who admit a*non*-empty Σ^1_1 subset. ... We need*to*extend their basis theorem by replacing*non*-*hyperarithmetic**sets*by compact*sets*with no*non*-empty Σ 1 1 subsets in order*to*prove the remaining direction of Theorem 2.1. ...##
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Measure-theoretic applications of higher Demuth's Theorem

2016
*
Transactions of the American Mathematical Society
*

We investigate measure-theoretic aspects of various notions of reducibility by applying analogs of Demuth's Theorem in the

doi:10.1090/tran/6881
fatcat:3pwzo3u7f5d5teqsna6idqsziu
*hyperarithmetic*and*set*-theoretic*settings*. ... The next theorem shows that*hyperarithmetic*reducibility*relative**to*a Π 1 1random x is in fact Turing reducibility*relative**to*the join of x with a*hyperarithmetic**set*. ... Relativizing Proposition 3.9*to*r, there are 2 ℵ 0 -many ω r 1 -generic reals {g γ } γ<2 ℵ 0 ,*any*two of which form an exact pair of hyperdegrees*relative**to*r. ...##
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The Slaman-Wehner theorem in higher recursion theory

2011
*
Proceedings of the American Mathematical Society
*

We note that in Theorem 1.1, Kleene's O cannot be replaced by

doi:10.1090/s0002-9939-2010-10693-7
fatcat:wlbvf2viufcdtl3m644q5afyzm
*any**hyperarithmetic**set*. ... Is there a countable structure M such that the collection of*sets*X ∈ 2 ω such that some copy of M is*hyperarithmetic*in X is exactly the collection of*non*-*hyperarithmetic**sets*? ...##
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The determined property of Baire in reverse math
[article]

2018
*
arXiv
*
pre-print

*Any*ω-model of DPB must be closed under

*hyperarithmetic*reduction, but DPB is not a theory of

*hyperarithmetic*analysis. ... We show that whenever M⊆ 2^ω is the second-order part of an ω-model of DPB, then for every Z ∈ M, there is a G ∈ M such that G is Δ^1_1-generic

*relative*

*to*Z. ... X is not generic

*relative*

*to*H Z b } After decorating this code, it becomes determined for every

*non*-∆ 1 1 (Z)-generic. ...

##
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The Strength of the Besicovitch-Davies Theorem
[chapter]

2010
*
Lecture Notes in Computer Science
*

We consider the weak (Muchnik) reducibility ≤w in connection with the mass problem S(U ) of computing a

doi:10.1007/978-3-642-13962-8_26
fatcat:3vm2icfh7neztcc4cbuqzcau3m
*set*X ⊆ ω such that the Σ 1 1 class U of positive dimension has a Π 0 1 (X) subclass of positive ... It was subsequently generalized*to*various*non*-Euclidean*settings*. ... The hierarchies of effective descriptive*set*theory allow for a further ramification of regularity properties.*Any*(boldface) Borel*set*is effectively (lightface) Borel*relative**to*a parameter. ...##
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Categoricity of computable infinitary theories

2008
*
Archive for Mathematical Logic
*

Suppose the following conditions are satisfied: 1. for

doi:10.1007/s00153-008-0117-z
fatcat:p3fy5mjkn5fglaw3myfqcarqyy
*any*choice functions c and c ,*any*isomorphism f from A c onto A c extends*to*an automorphism of B, 2. for*any*tuple b in B and*any*choice function ... We have the following general result for this*setting*. Transfer Theorem I. Let B be a*hyperarithmetical*structure. ... Let A and B be*hyperarithmetical*structures. Let X be a definable subset of B, and let g be a*hyperarithmetical*function from X onto the*set*of tuples in A s.t. the following conditions are satisfied: ...##
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Page 4609 of Mathematical Reviews Vol. , Issue 2000g
[page]

2000
*
Mathematical Reviews
*

permutations as well as that

*any**hyperarithmetical*function is a product of algebraic elements over recursive functions.” ... of*relative*computability. ...##
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Page 742 of Mathematical Reviews Vol. 35, Issue 4
[page]

1968
*
Mathematical Reviews
*

*set*is the union of two disjoint

*non*-

*hyperarithmetic*I1,*

*sets*. The proofs of these two theorems are not difficult

*to*reconstruct. ... In § 5 of the present paper, the authors confine | themselves

*to*announcing: (i) There exists a maximal II,'

*set*; (ii) every

*non*-

*hyperarithmetic*[1,? ...

##
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Computability Theory, Nonstandard Analysis, and their connections
[article]

2017
*
arXiv
*
pre-print

(T.1) A basic property of Cantor space 2^N is Heine-Borel compactness: For

arXiv:1702.06556v2
fatcat:6l7wcxys7jddjk46vfjsctk23y
*any*open cover of 2^N, there is a finite sub-cover. ... The study of (T.1) gives rise*to*exotic objects in computability theory, while (T.2) leads*to*surprising results in Reverse Mathematics. ... Now observe that {f α h(f α ) : α ≤ α h } is definable as the closure*set*of a nonmonotonic arithmetical inductive definition*relative**to*h, so this*set*will have complexity ∆ 1 2*relative**to*h. ...##
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Measures and their random reals
[article]

2013
*
arXiv
*
pre-print

On the other hand, examples of reals not random for

arXiv:0802.2705v2
fatcat:pxa56lgcoja5fckkgrd3hjruum
*any*continuous measure can be found throughout the*hyperarithmetical*Turing degrees. ... We show that every*non*-computable real is*non*-trivially random with respect*to*some measure. The probability measures constructed in the proof may have atoms. ... For*any*z ∈ 2 ω , the following are equivalent. (i) x is random*relative**to*z for a continuous measure µ recursive in z. ...
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