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Nonbounding and Slaman triples

Steven D. Leonhardi
<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bnojym2hjzcgnpa4wixi2axhnq" style="color: black;">Annals of Pure and Applied Logic</a> </i> &nbsp;
The existence is shown of a high2 r.e. degree which does not hound what we call the base of any Slaman triple. *  ...  We consider the relationship of the lattice-theoretic properties and the jump-theoretic properties satisfied by a recursively enumerable Turing degree.  ...  Theorem 1.4 (Shore and Slaman [ 131) . Any high r.e. degree bounds some nonsplitting pair, and bounds some Slaman triple.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0168-0072(95)00044-5">doi:10.1016/0168-0072(95)00044-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fdohubhdlfbxhatl5dqvxcvpqi">fatcat:fdohubhdlfbxhatl5dqvxcvpqi</a> </span>
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Page 4092 of Mathematical Reviews Vol. , Issue 98G [page]

<span title="">1998</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Downey (NZ-VCTR; Wellington) 98g:03102 03D25 Leonhardi, Steven D. (1-MRVN; Bethlehem, PA) Nonbounding and Slaman triples. (English summary) Ann. Pure Appl. Logic 79 (1996), no. 2, 139-163.  ...  The degrees u,v,w form a Slaman triple with base u if u is not recursive, w £ v, and w < bUv whenever 0 <b < u.  ... 
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Intervals Without Critical Triples [chapter]

Peter Cholak, Rod Downey, Richard Shore
<span title="">1998</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/gwgjc2cgezbb3o5utsihhe7yze" style="color: black;">Logic Colloquium &#39;95</a> </i> &nbsp;
Actually, w e construct intervals I of computably enumerable degrees without any w eak critical triples (this implies that M 5 cannot be embedded in I, see Section 2).  ...  Our strongest result is that there is a low 2 computably enumerable degree e such that there are no weak critical triples in either of the intervals 0 e] o r e 0 0 ]. 1991 Mathematics Subject Classi cation  ...  Upward and downward nonbounding. Theorem 6.6. There i s a c omputably enumerable degree e above which there i s n o w e ak critical triple and below which there i s n o w e ak critical triple.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-662-22108-2_2">doi:10.1007/978-3-662-22108-2_2</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/logicColl/CholakDS95.html">dblp:conf/logicColl/CholakDS95</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bq2bw4kibbaxhitnoaekqludge">fatcat:bq2bw4kibbaxhitnoaekqludge</a> </span>
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NONSTANDARD MODELS IN RECURSION THEORY AND REVERSE MATHEMATICS

C. T. CHONG, WEI LI, YUE YANG
<span title="">2014</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nxnim3i24fhcvkvgruk2dej7yu" style="color: black;">Bulletin of Symbolic Logic</a> </i> &nbsp;
We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic  ...  AbstractWe give a survey of the study of nonstandard models in recursion theory and reverse mathematics.  ...  Is Lachlan's nonsplitting theorem [36] or nonbounding theorem [37] equivalent to IΣ 3 over the base theory P − + BΣ 3 ? The same question applies to the existence of a Slaman triple [55] .  ... 
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Degrees of Unsolvability [chapter]

Klaus Ambos-Spies, Peter A. Fejer
<span title="">2014</span> <i title="Elsevier"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3htcvojut5de5ct6zf42o424vi" style="color: black;">Handbook of the History of Logic</a> </i> &nbsp;
We finish our history by quoting Gerald Sacks, who through his own work and the work of his students (including, Harrington, Robinson, Shore, Simpson, Slaman, and Thomason) shaped degree theory throughout  ...  The proof is based on Lachlan's nonbounding technique and uses contiguous degrees for creating local distributivity.  ...  In fact, he has shown that for elementarily equivalent upper cones, the bases must have the same triple jump.  ... 
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On the distribution of Lachlan nonsplitting bases

S. Barry Cooper, Angsheng Li, Xiaoding Yi
<span title="2002-07-01">2002</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3egevcnserbbni4tcigsztdici" style="color: black;">Archive for Mathematical Logic</a> </i> &nbsp;
the relationship between bounding and nonbounding of Lachlan nonsplitting bases and the high /low hierarchy.  ...  We say that a computably enumerable (c.e.) degree b is a Lachlan nonsplitting base (LNB), if there is a computably enumerable degree a such that a > b, and for any c.e. degrees In this paper we investigate  ...  Leonhardi [1996] proved that there is a high 2 c.e. degree which bounds no Slaman triple, and noted the difference between bounding the Slaman triple and bounding the Lachlan nonsplitting base, asking  ... 
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Page 104 of Mathematical Reviews Vol. , Issue Index [page]

<i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Nonbounding and Slaman triples. (English summary) 98g:03102 — Generalized nonsplitting in the recursively enumerable degrees.  ...  Russian summary) 98¢:03090 Lempp, Steffen (with Lerman, Manuel) Iterated trees of strategies and priority arguments. (English summary) 98m:03086 — (with Nies, André; Slaman, Theodore A.)  ... 
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PARAMETER DEFINABILITY IN THE RECURSIVELY ENUMERABLE DEGREES

ANDRÉ NIES
<span title="">2003</span> <i title="World Scientific Pub Co Pte Lt"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hsmcio5afvhcpaz6kfjsdapfqu" style="color: black;">Journal of Mathematical Logic</a> </i> &nbsp;
As applications, we show that Low 1 is parameter definable, and we provide a new example of a ∅-definable ideal.  ...  Shore and T. Slaman for many helpful discussions. He incorporated ideas from work of Shore and Slaman (unpublished) , who previously had obtained the main result in case d is noncappable.  ...  . , T n have higher priority than M i , and (W i , Φ i , Θ i ) is an effective list of all triples consisting of a r.e. set W i and two functionals Φ i , Θ i , such that W i ∩ 2N = E .  ... 
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Lattice embeddings below a nonlow2 recursively enumerable degree

Rod Downey, Richard A. Shore
<span title="">1996</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/whoel4vntnhddbjlrtj3w2jgra" style="color: black;">Israel Journal of Mathematics</a> </i> &nbsp;
triple either above or below a.  ...  It might seem reasonable to suggest that if a < b and b "sufficiently high over a"one should be able to embed a critical triple in [a, b].  ... 
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Non-bounding constructions

J.R. Shoenfield
<span title="">1990</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bnojym2hjzcgnpa4wixi2axhnq" style="color: black;">Annals of Pure and Applied Logic</a> </i> &nbsp;
The key ideas in this common part are due to Lachlan; we include several improvements due to Harrington, Soare, Slaman, and the author. Our notation is fairly standard.  ...  The object of this paper is to explain a certain type of construction which occurs in priority proofs and illustrate it with two examples due to Lachlan and Harrington.  ...  Rather surprisingly, one rule suffices for nonbounding constructions. We say that do is i-good at s if either L,(i) 2 N,(a), or there is a t such that N,(a) G t =SS and L,(i) c f(a). Rule 5.  ... 
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Intervals without Critical Triples [chapter]

P. Cholak, R. Downey, R. Shore, Johann A. Makowsky, Elena V. Ravve
<i title="Cambridge University Press"> Logic Colloquium &#39;95 </i> &nbsp;
Actually, w e construct intervals I of computably enumerable degrees without any w eak critical triples (this implies that M 5 cannot be embedded in I, see Section 2).  ...  Our strongest result is that there is a low 2 computably enumerable degree e such that there are no weak critical triples in either of the intervals 0 e] o r e 0 0 ]. 1991 Mathematics Subject Classi cation  ...  Upward and downward nonbounding. Theorem 6.6. There i s a c omputably enumerable degree e above which there i s n o w e ak critical triple and below which there i s n o w e ak critical triple.  ... 
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On Lachlan's major sub-degree problem

S. Barry Cooper, Angsheng Li
<span title="2008-06-27">2008</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3egevcnserbbni4tcigsztdici" style="color: black;">Archive for Mathematical Logic</a> </i> &nbsp;
Harrington showed (see Shore and Slaman [1990] ): 1.16 THEOREM.  ...  We say that d(k) receives honestification at stage s via B-change, and go back to step 2. [v] + 1), then set Γ(X, A; k) to be undefined, and cancel d(k ′ ) for all k ′ > k.  ...  By the proof of Lemmas 8.15 and 8.16, there is a stage s 2 > s 1 such that no strategy β with α ⊂ β and αˆ 2 ⊆ β acts at any odd stage s > s 2 .  ... 
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