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Page 8388 of Mathematical Reviews Vol. , Issue 2000m
[page]

2000
*
Mathematical Reviews
*

In fact, Shore and Slaman extend the Jockusch and Shore result to a-

*REA**operators*and the a-*jump*,*for*all a. ... Shore and Slaman have shown that y(a) fails*for*all n-*REA*degrees a [“A splitting*theorem**for*n-*REA*degrees”, Proc. Amer. Math. ...##
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Page 637 of Mathematical Reviews Vol. , Issue 95b
[page]

1995
*
Mathematical Reviews
*

Analogues of the Sacks

*jump**theorem*are also obtained*for*n-fold and wn-fold iterations of*REA**operators*. The proofs involve clever applications of the recursion*theorem*. C. G. ... Shore extended a number of results about the Turing*jump**operator*to*REA**operators*, i.e. those of the form J,(A) = A@ WA*for*some e [see, e.g., J. ...##
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Page 5415 of Mathematical Reviews Vol. , Issue 87j
[page]

1987
*
Mathematical Reviews
*

The

*jump**operator*is an*REA**operator*, up to recursive isomorphism. Existence proofs in the theory of r.e. degrees give many more examples of*REA**operators*. ... An*operator*on “2 of the form J,*for*some e is called an*REA**operator*or, sometimes, a pseudojump*operator*. (Here ‘*REA*’ stands*for*‘r.e. in and above’.) ...##
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Degree Structures: Local and Global Investigations

2006
*
Bulletin of Symbolic Logic
*

Institutionally, it was an honor to serve as President of the Association and I want to thank my teachers and predecessors

doi:10.2178/bsl/1154698739
fatcat:ct6b4ko77zhyfctmtfooxbk4ua
*for*guidance and advice and my fellow officers and our publisher*for*their work ... Appropriately enough, my first administrative job*for*the Association, some thirty years ago, was to serve on a committee to plan a reorganization of the reviews. ... Together with the join*theorem**for*n-*REA**operators*it gives the following de…nition of the Turing*jump*from that of the double*jump*..*Theorem*4.16. ...##
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The jump is definable in the structure of the degrees of unsolvability

1990
*
Bulletin of the American Mathematical Society
*

Kleene and Post also considered the enriched structure 3 1 equipped with the "

doi:10.1090/s0273-0979-1990-15923-3
fatcat:d3g7bx5i6neufhcr5b5i32okie
*jump**operator*", denoted ', which is a canonical*operation*on degrees which takes each degree d to a strictly ... Richter [24] showed that each*jump*preserving automorphism of the degrees fixes all a > 0 (3) . • Although the join*theorem*is not known*for*arbitrary*REA**operators*, we can apply the special cupping ... derivable*for**operators*of the particular form got from H-r.e.*operators*. ...##
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Defining jump classes in the degrees below ${\bf 0}'$

1988
*
Proceedings of the American Mathematical Society
*

Thus

doi:10.1090/s0002-9939-1988-0958085-4
fatcat:ktjqutmsxvasfeqphy6sbt3p6u
*for*each r.e. degree r, Th(^(< r)) uniquely determines r(3). ... We prove that,*for*each degree c r.e. in and above O'3', the class of degrees x < 0' with x'3) = c is definable without parameters in 3{< 0'), the degrees below 0'. ... Defining the*jump*classes.*THEOREM*2.1.*For*each r.e. degree r and each c*rea*in 0'3^ the class of degrees x < r such that x.^ = c is definable in 3(< r). PROOF. ...##
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Defining the Turing Jump

1999
*
Mathematical Research Letters
*

Now

doi:10.4310/mrl.1999.v6.n6.a10
fatcat:hmel7rrvw5bcvokljzcjytw6yq
*for*the completeness*theorem*needed.*Theorem*1.6 (Jockusch and Shore (1984) )*For*any α−*REA**operator*J and any C ≥ T 0 (α) there is an A such that J(A) ≡ T C. ... n-*REA**Operators*and Kumabe-Slaman Forcing n-*REA**Operators*and Kumabe-Slaman Forcing n-*REA**Operators*and Kumabe-Slaman Forcing n-*REA**Operators*and Kumabe-Slaman Forcing n-*REA**Operators*... The uniformity of this construction clearly provides a proof of the*theorem**for*the ω −*REA**operators*as well. ...##
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Local Definitions in Degree Structures: The Turing Jump, Hyperdegrees and Beyond

2007
*
Bulletin of Symbolic Logic
*

Analogous results hold

doi:10.2178/bsl/1185803806
fatcat:6s4wggdofjdrzbk4op7yr27t5q
*for*various coarser degree structures. ... There are Π5 formulas in the language of the Turing degrees, D, with ≤, ⋁ and ⋀, that define the relations x″ ≤ y″, x″ = y″ and so x ∈ L 2(y) = {x ≥ y ∣ x″ = y″} in any*jump*ideal containing 0 (ω). ... The join*theorem**for*n-*REA**operators*of Shore and Slaman [1999] then de…nes the Turing*jump*from that of the double*jump*:*For*any degree x, x 0 = maxfz T xj(8g T x)(z _ g 6 = g 00 )g, i.e. x 0 is the ...##
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On the Jumps of the Degrees Below a Recursively Enumerable Degree

2018
*
Notre Dame Journal of Formal Logic
*

The implication (1 ⇒ 3) follows from the monotonicity of the

doi:10.1215/00294527-2017-0014
fatcat:qmzj4i55ofgytjpoy4vajcr7iu
*jump**operator*and the fact that if X ≤ T A and Y is r.e. in X then Y is r.e. in A. The final implication (3 ⇒ 2) follows from*Theorem*1.2. ... . * Both authors were partially supported by NSF Grant DMS-1161175. 1. y = x*for*some x ≤ a. 2. y = x*for*some r.e. x ≤ a. 3. y ≥ 0 and y is*REA*(a). y = deg(W Proof. ...*Theorem*1.2 (Robinson*Jump*Interpolation) . If c, d, e ∈ R, e c < d, z ≥ c and is*REA*(d), then there is an f ∈ R with c < f < d, e f , and f = z. ...##
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DIRECT AND LOCAL DEFINITIONS OF THE TURING JUMP

2007
*
Journal of Mathematical Logic
*

*For*a lower bound on de...nability, we show that no 2 or 2 formula in the language with just de...nes L 2 or L 2 (y). ... We show that there are 5 formulas in the language of the Turing degrees, D, with ,_ and^, that de...ne the relations x 00 y 00 , x 00 = y 00 and so x 2 L 2 (y) = fx yjx 00 = y 00 g in any

*jump*ideal containing ... The join

*theorem*

*for*n-

*REA*

*operators*of Shore and Slaman [1999] then de…nes the Turing

*jump*from that of the double

*jump*:

*For*any degree x, x 0 = maxfz T xj(8g T x)(z _ g 6 = g 00 )g, i.e. x 0 is the ...

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Page 578 of Mathematical Reviews Vol. , Issue 93b
[page]

1993
*
Mathematical Reviews
*

The authors prove, via a forcing argument, that every countable

*jump*partial ordering is embeddable in (D, <r, jy), the Turing degrees with the*jump**operator*. ... Another observation shows that, in fact, the e-degrees of the n-*rea*sets are dense*for*each n. ...##
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Abstract hierarchies and degrees

1989
*
Journal of Symbolic Logic (JSL)
*

*Theorem*3 . 3 11.

*For*all a there is a degree a' called the

*jump*of a s.t. a' ~

*REa*and whenever b ~

*REa*, then b ~a'. ... The following statement introduces a

*jump*

*operator*over D. Proof: Take a ~ E a, then take a' = dgi: (~+) by definition. It is immediate that a' < a. Suppoc;e that b ~RE a. ...

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Low Level Nondelegability Results: Domination and Recursive Enumeration

2013
*
Journal of Symbolic Logic (JSL)
*

We prove a variety of results, including,

doi:10.2178/jsl.7803180
fatcat:c26gbzzhfzahvde4pyt2g2yj3m
*for*example, that being array nonrecursive is not definable by a Σ1 or Π1 formula in the language (≤,*REA*) where*REA*stands*for*the "r.e. in and above" predicate ... We also show that the Σ1-theory of (, ≤,*REA*) is decidable. ... In addition to the basic language ( ), they suggest two extensions augmenting it by either the*jump**operator*0 or the relation RE*for*"recursively enumerable in." ...##
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Page 6730 of Mathematical Reviews Vol. , Issue 95k
[page]

1995
*
Mathematical Reviews
*

95k:47104 47

*OPERATOR*THEORY 6730 ity of the*operator*I’ there is an incomprehensible*jump*from the modulars py and py to py, and py, (p. 741, starting at line 6). (6) In the formulation of*Theorems*1, ... +a, -1<*Rea*< 0}, which is bounded by the “critical parabola” (4) P = {A¢€ C: 4=a*?+a,*Rea*=0}. ...##
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Interpolating d-r.e. and REA degrees between r.e. degrees

1996
*
Annals of Pure and Applied Logic
*

Here the motivating ideas were the

doi:10.1016/0168-0072(96)00014-0
fatcat:7lxj22cs6nelfkblxlqdwklfxm
*jump**operator*and relative recursive enumerability. Definition 1.2. We define the sets*REA*in X by induction. 1. X is 0-*REA*in X. 2. ... ., c is low and h is high, then there is an a < h which is*REA*in c but not r.e.*Theorem*2.1. ... Then a generalization of both the Friedberg completeness*theorem*and the Posner-Robinson cupping*theorem**for*aREA*operators*derived from a-r.e. ones proved in [22] is applied to see that every X 6 O( ...
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