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Relative to any non-hyperarithmetic set [article]

Noam Greenberg, Antonio Montalban, Theodore Slaman
2011 arXiv   pre-print
We prove that there is a structure, indeed a linear ordering, whose degree spectrum is the 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.  ... 
arXiv:1110.1907v1 fatcat:v2qnzyoxmnekliihwqjy5kqfni

Analytic equivalence relations satisfying hyperarithmetic-is-recursive [article]

Antonio Montalbán
2013 arXiv   pre-print
We prove, in ZF+Σ^1_2-determinacy, that for 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.  ... 
arXiv:1306.1513v2 fatcat:zvqknsr255gjrjcxgnctajvymu

Finding subsets of positive measure [article]

Bjørn Kjos-Hanssen, Jan Reimann
2014 arXiv   pre-print
On the other hand, there are Π^0_2 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.  ... 
arXiv:1408.1999v1 fatcat:zi4s6nw2azatvapyheo7nab6uy

Computable structures of rank omega_1^CK [article]

Julia Knight, Jessica Millar
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 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.  ... 
arXiv:math/0508507v1 fatcat:bu3gyfj6s5he7enhji37hkknry

Equivalence Relations on Classes of Computable Structures [chapter]

Ekaterina B. Fokina, Sy-David Friedman
2009 Lecture Notes in Computer Science  
If the index 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.  ... 
doi:10.1007/978-3-642-03073-4_21 fatcat:tkc2imruzrc2zg44ygekmrtoe4

Pi01 encodability and omniscient reductions [article]

Benoit Monin, Ludovic Patey
2016 arXiv   pre-print
By a result of Solovay, the computably encodable 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.  ... 
arXiv:1603.01086v2 fatcat:brz5d6tobnaqlcy5e62u6hrdyq

Measure-theoretic applications of higher Demuth's Theorem

C. T. Chong, Liang Yu
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 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.  ... 
doi:10.1090/tran/6881 fatcat:3pwzo3u7f5d5teqsna6idqsziu

The Slaman-Wehner theorem in higher recursion theory

Noam Greenberg, Antonio Montalbán, Theodore A. Slaman
2011 Proceedings of the American Mathematical Society  
We note that in Theorem 1.1, Kleene's O cannot be replaced by 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?  ... 
doi:10.1090/s0002-9939-2010-10693-7 fatcat:wlbvf2viufcdtl3m644q5afyzm

The determined property of Baire in reverse math [article]

Eric P. Astor, Damir Dzhafarov, Antonio Montalbán, Reed Solomon, Linda Brown Westrick
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.  ... 
arXiv:1809.03940v2 fatcat:gcdbwftcwvffxcxrj6haz3xypa

The Strength of the Besicovitch-Davies Theorem [chapter]

Bjørn Kjos-Hanssen, Jan Reimann
2010 Lecture Notes in Computer Science  
We consider the weak (Muchnik) reducibility ≤w in connection with the mass problem S(U ) of computing a 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.  ... 
doi:10.1007/978-3-642-13962-8_26 fatcat:3vm2icfh7neztcc4cbuqzcau3m

Categoricity of computable infinitary theories

W. Calvert, S. S. Goncharov, J. F. Knight, Jessica Millar
2008 Archive for Mathematical Logic  
Suppose the following conditions are satisfied: 1. for 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:  ... 
doi:10.1007/s00153-008-0117-z fatcat:p3fy5mjkn5fglaw3myfqcarqyy

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.  ... 

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,?  ... 

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

Dag Normann, Sam Sanders
2017 arXiv   pre-print
(T.1) A basic property of Cantor space 2^N is Heine-Borel compactness: For 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.  ... 
arXiv:1702.06556v2 fatcat:6l7wcxys7jddjk46vfjsctk23y

Measures and their random reals [article]

Jan Reimann, Theodore A. Slaman
2013 arXiv   pre-print
On the other hand, examples of reals not random for 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.  ... 
arXiv:0802.2705v2 fatcat:pxa56lgcoja5fckkgrd3hjruum
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