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A non-splitting theorem for d.r.e. sets

1996
*
Annals of Pure and Applied Logic
*

Sacks showed that

doi:10.1016/0168-0072(95)00070-4
fatcat:enkgzixxwreqbawb7n66q3a3fq
*for*each*non*-recursive recursively enumerable*set**A*there are disjoint recursively enumerable*sets*B, C which cover*A*such that*A*is recursive in neither*A*n B nor*A*n C. ... Abstract*A**set*of natural numbers is called*d.r.e*. ... There exists*a*properly*d.r.e*.*set*D such that*for*all r.e.*sets*AO, A1 with A0 n A1 = 0, D~*A*O U*A*~~[ D <~*A*O~D V D <~*A*~~D ] . r 6 3 ' 3 J,*for*all j E {jo,. ...##
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Page 5961 of Mathematical Reviews Vol. , Issue 97J
[page]

1997
*
Mathematical Reviews
*

This is in contrast to Sacks’s

*splitting**theorem**for*re.*sets*. ... Peter Cholak (1-NDM; Notre Dame, IN) 03D Recursion theory 97}:03089 97j:03086 03D30 03D25 Yi, Xiaoding (1-CT; Storrs, CT)*A**non*-*splitting**theorem**for**d.r.e*.*sets*. (English summary) Ann. Pure Appl. ...##
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Recursively enumerable sets and degrees

1978
*
Bulletin of the American Mathematical Society
*

The structure, automorphisms, and elementary theory of the r.e.

doi:10.1090/s0002-9904-1978-14552-2
fatcat:arxp4btvhzfzjeoufybmnemt2u
*sets*. 6. Basic facts and*splitting**theorems*. 7. Hh-simple*sets*. 8. Major subsets and r-maximal*sets*. 9. Automorphisms of &. 10. ...*A*subset*A*of <o (the*set*of nonnegative integers) is recursive (also called decidable or computable) if there is an algorithm*for*determining whether*a*...*SETS*6. Basic facts and*splitting**theorems*. ...##
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Master index to volumes 61-70

1994
*
Annals of Pure and Applied Logic
*

Jin, R. and Shelah, S., Essential Kurepa trees versus essential Jech-Kunen trees Joyal,

doi:10.1016/s0168-0072(94)90013-2
fatcat:jcewrks7xbf4peme3526yoypre
*A*. and Moerdijk, I.,*A*completeness*theorem**for*open maps Kaddah, D., Infima in the*d.r.e*. degrees Kagan, V., Nerode ... with*non*-constructive p-operator. ...##
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d-simple sets, small sets, and degree classes

1980
*
Pacific Journal of Mathematics
*

Thus ^iCΰ and D

doi:10.2140/pjm.1980.87.135
fatcat:xilieu56zbh7fi57s2wg3bbyxa
*splits*L u and likewise*for*D replaced by N, the degrees containing*non*-small*sets*. We do not know whether D^L λ or whether N -D.*THEOREM*4.1. ... The construction is very similar to the usual construction [18,*Theorem*4.1] of*a*low simple*set**A*except that*A*must now intersect certain infinite*d.r.e*.*sets*instead of certain infinite r.e.*sets*. ...##
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The index set of uncountably categorical theories

2013
*
Israel Journal of Mathematics
*

We show that this index

doi:10.1007/s11856-013-0011-1
fatcat:3ficgnd3v5dy5lq3siqgy7miwi
*set*surprisingly falls at the intermediate stage of being complete*for*intersections of Π 2*sets*with Σ 2*sets*. ... We classify the complexity of the index*set*of uncountably categorical theories. ... Surprisingly, uncountable categoricity is arithmetical and lies at an intermediate level, being complete*for*intersections of*a*Π 2*set*and*a*Σ 2*set*(also known as 0 -*d.r.e*.*sets*). ...##
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On α- and β-recursively enumerable degrees

1979
*
Annals of Mathematical Logic
*

In particular by applying the

doi:10.1016/0003-4843(79)90002-0
fatcat:kaew2ei5gjgb7p4g76ba277uly
*splitting**theorem*to many local structures we get*a*global*splitting**theorem**for*all regular /3-r.e. degrees. ... It turns out that routine precautions are sufficient to make the proof work*for*all admissible structures. Call*a**set*D 9l-*d.r.e*. if D =*A*-B*for*some ')l-r e.*sets**A*. B. ...##
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On the Turing Degrees of Weakly Computable Real Numbers

2003
*
Journal of Logic and Computation
*

The Turing degree of

doi:10.1093/logcom/13.2.159
fatcat:4lp67kl5vvh2tpphr53gxt4w6e
*a*real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. ... Therefore, every*non*-computable c.e. degree contains both semi-computable and*non*-semi-computable real numbers.*Theorem*1.7*For*any*non*-computable c.e. ... By Sacks'*Splitting**Theorem*[14] there exist two incomparable c.e. degrees b 0 , b 1 such that*a*= b 0 ∪ b 1 . Choose two c.e.*sets*B 0 ∈ b 0 and B 1 ∈ b 1 and define*set**A*=: B 0 ⊕ B 1 . ...##
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Page 96 of Mathematical Reviews Vol. , Issue Index
[page]

*
Mathematical Reviews
*

*theorem*

*for*

*d.r.e*.

*sets*. ... (English summary) 971:03026 — (with Sui, Yuefei) An extended Lachlan

*splitting*

*theorem*. (English summary) 97b:03055 — see also Arsianov, M. ...

##
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Page 104 of Mathematical Reviews Vol. , Issue Index
[page]

*
Mathematical Reviews
*

(English summary) 98c:03092
Kontostathis, Kyriakos The combinatorics of the

*splitting**theorem*. 98j:03059 Kumabe, Masahiro Minimal complementation below uniform upper bounds*for*the arithmetical degrees ... 03D25 03D25 Recursively enumerable*sets*and degrees Cichon, E.*A*. (with Weiermann, Andreas) Term rewriting theory*for*the primitive recursive functions. (English summary) 98b:03057 Cooper, S. ...##
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2015 EUROPEAN SUMMER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC LOGIC COLLOQUIUM '15 Helsinki, Finland August 3–8, 2015

2016
*
Bulletin of Symbolic Logic
*

, or

doi:10.1017/bsl.2016.22
fatcat:cm4a5yyvgfajzm4qvlszviblni
*a**set*of assignments. ... In*set*theory,*a*reflection principle says that*a*proposition true in the universe holds already in*a*smaller*set*. ... We prove that*for*some sequence of tautologies ϕn the proof steps and the proof sizes in ...##
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Page 110 of Mathematical Reviews Vol. 32, Issue Index
[page]

*
Mathematical Reviews
*

Maximal

*sets*and the jump operator. 2000a:03065 Ding, Decheng see Lu, Hong et al., 2000f:03123 and 2000m:03100 Downey, Rodney G. (with Shore, Richard*A*.)*Splitting**theorems*and the jump operator. ... Arslanov, Asat Difference*splittings*of recursively enumerable*sets*. ...##
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Page 90 of Mathematical Reviews Vol. 27, Issue Index
[page]

*
Mathematical Reviews
*

(English summary) 95h:03103

*Splitting**theorems*in recursion theory. (English Highness and bounding minimal pairs. with Kummer, Martin Walter) Diagonals and ®-maximal*sets*. ... (with Yi, Xiaoding) Jump*theorems**for*REA operators. (English summary) 95b:03048 Morales-Luna, Guillermo Degrees of complexity of recursive functions and degrees of unsolvability. (Spanish. ...##
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Page 1638 of Mathematical Reviews Vol. , Issue Index
[page]

*
Mathematical Reviews
*

(Peter Cholak) 97e:03063 03D25
—

*A**non*-*splitting**theorem**for**d.r.e*.*sets*. (English summary) Ann. Pure Appl. Logic 82 (1996), no. 1, 17-96. ... (see 97a:90003) 93E20 (90B70, 90C40) — (with Zhang, Qing')*A*central limit*theorem**for*singularly perturbed nonstationary finite state Markov chains. (English summary) Ann. Appl. ...