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Metarecursively enumerable sets and admissible ordinals
1966
Bulletin of the American Mathematical Society
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It was observed in [8] that there exist bounded, metarecursively enumerable sets which are not metarecursive; each such set is a constructive example of a nonregular set. ...
A set of recursive ordinals is called regular if its intersection with every metafinite set of recursive ordinals is metafinite. (The metafinite sets coincide with the bounded, metarecursive sets.) ...
Each metarecursively enumerable set has the same metadegree as some regular, metarecursively enumerable set. Two sets have the same metadegree [8] if each is metarecursive in the other. ...
doi:10.1090/s0002-9904-1966-11416-7
fatcat:howe4vmtibf45dtvilxup5ktce
Post's problem, admissible ordinals, and regularity
1966
Transactions of the American Mathematical Society
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Theorem 4 of [7] states that there exist two metarecursively enumerable sets of recursive ordinals such that neither is metarecursive in the other. ...
Unfortunately, there exists a multitude of nonregular, metarecursively enumerable sets [7] . ...
Each metarecursively enumerable set has the same metadegree as some regular, metarecursively enumerable set. Proof. ...
doi:10.1090/s0002-9947-1966-0201299-1
fatcat:etht5unbhvajnitfuehznm6aia
Page 742 of Mathematical Reviews Vol. 35, Issue 4
[page]
1968
Mathematical Reviews
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Spector [loc. cit.] showed that all I1,* sets lie in two comparable hyper- degrees, so that metarecursion theory will clearly be of more help in exploring the fine structure of II, sets. ...
Finally, a simple but important observation made by the authors is that there are bounded meta r.e. sets which are not metarecursive: take, for example, any II, set of unique notations for the recursive ...
On the reducibility of ⊓11 sets
1971
Advances in Mathematics
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A set is metafinite if it is metarecursive and bounded (by some ordinal < wi). There exists a metarecursive indexing of the metafinite sets. ...
A set A is metarecursive if its characteristic function is metarecursive, or equivalently, if both A and wi -A are metarecursively enumerable. ...
doi:10.1016/0001-8708(71)90042-9
fatcat:rmf5tkwiy5ewxnfevbwlbl7dxu
Page 16 of Mathematical Reviews Vol. 44, Issue 1
[page]
1972
Mathematical Reviews
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to metarecursion theory and proves that there is a metarecursively enumerable sequence S(a), «<w,, such that to each metarecursively enumerable set W there corresponds a unique ordinal a for which W =S ...
However, there exists a metarecursive enumeration S(a), «<w,, of all infinite I1,* sets. R. L. Goodstein (Leicester)
Schwichtenberg, Helmut 77
Eine Klassifikation der ¢,-rekursiven Funktionen.
Z. ...
Page 1289 of Mathematical Reviews Vol. 40, Issue 6
[page]
1970
Mathematical Reviews
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Metarecursion theory. Sets, Models and Recursion Theory (Proc. Summer School Math. Logic and Tenth Logic Colloq., Leicester, 1965), pp. 243-263. North-Holland, Amsterdam, 1967. ...
Throughout the author stresses distinct notions which have different extensions in metarecursion theory, but coincide in the classical case, mainly concerning reducibility between meta-r.e. sets. ...
Page 11 of Mathematical Reviews Vol. 43, Issue 1
[page]
1972
Mathematical Reviews
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Driscoll, Graham C., Jr. 46 Metarecursively enumerable sets and their metadegrees. J. Symbolic Logic 38 (1968), 389-411.
This is an important paper in metarecursion theory, intro-
duced by G. ...
One would like to define a set of recursive ordinals A to be metarecursive in a set of recursive ordinals B just in case each question about membership in A can be answered using only metafinitely much ...
The Role of True Finiteness in the Admissible Recursively Enumerable Degrees
2005
Bulletin of Symbolic Logic
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The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natural elementary differences between the structures ...
These notions coincide with those of metarecursion theory when α = ω CK 1 , which is the least admissible ordinal. ...
First metarecursion theory (Sacks [30] ) and then α-recursion theory (Sacks and Simpson [28] ) passed this test. ...
doi:10.2178/bsl/1122038994
fatcat:5h7jio3leza53nvscnmymv2yum
The role of true finiteness in the admissible recursively enumerable degrees
2006
Memoirs of the American Mathematical Society
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The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natural elementary differences between the structures ...
These notions coincide with those of metarecursion theory when α = ω CK 1 , which is the least admissible ordinal. ...
First metarecursion theory (Sacks [30] ) and then α-recursion theory (Sacks and Simpson [28] ) passed this test. ...
doi:10.1090/memo/0854
fatcat:kxkbxy6jwbavxo5asuffdfgnom
Page 44 of Mathematical Reviews Vol. , Issue 90A
[page]
1990
Mathematical Reviews
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For the class of autonomous numerations we establish their metarecursiveness.”
03E Set theory
See also 01057, «03037, 03063, 03067, 03110, 18006, 20009, 20141, 26037, 28032, 28033, 46009, 46047, 54011, ...
Metarecursiveness of autonomous numerations. (Russian)
Vychisl. Sistemy No. 122 Prikl. Aspekty Mat. Logiki (1987), 145-156, 166. ...
Recursiveness in $P^1_1$ paths through $\mathcal{O}$
1976
Proceedings of the American Mathematical Society
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If every hyp set is recursive in a given nj set, then 0 is recursive in its triple jump. Let 0 and <e be defined as in Rogers [7, p. 208]. ...
The proof proceeds informally, using concepts from metarecursion Received by the editors by the editors May 22, 1974 and, in revised form, February 18, 1975 ...
Since A,/are metarecursive, it is clear that <a"> is metarecursive. Hence lim(a") = X < coi. Since lim(0(/(a"))) = 0(/(A)), it is easy to see that F(e(f(X))) = A, and we are done. ...
doi:10.1090/s0002-9939-1976-0398812-2
fatcat:cw4d3xwmkjhxbop2fhpveyfn3q
Page 35 of Mathematical Reviews Vol. , Issue 92a
[page]
1992
Mathematical Reviews
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In Part B, metarecursion, the proper general- ization of recursion theory to hyperarithmetic theory, is introduced and discussed. ...
Ambos-Spies [same journal 31 (1985), no. 5, 461-477; MR 87d:03113] initiated the study of splitting of r.e. sets by build- ing an r.e. set A such that for any splitting into r.e. sets Ap and A), Ao and ...
JSL volume 32 issue 1 Cover and Front matter
1967
Journal of Symbolic Logic (JSL)
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By MARIKO YASUGI 145 Simplicity of recursively enumerable sets. By ROBERT W. ROBINSON . . . 162 Recursion, metarecursion, and inclusion. By JAMES C. ...
There exist two regressive sets whose intersection is not regressive. By K. I. ...
doi:10.1017/s0022481200114501
fatcat:zezl2axhirfjlbi2xi5fhanqb4
Abstracts of papers
1966
Journal of Symbolic Logic (JSL)
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Some applications of metarecursion theory to IL\-sets. G. E. ...
Driscoll proves that weak relative metarecursiveness (written g w ) is not a transitive relation on the meta-r.e. sets (see Metarecursive sets for definitions). ...
doi:10.1017/s0022481200126258
fatcat:y5dyvpw3cffcxikdbyhmmciyou
Page 4135 of Mathematical Reviews Vol. , Issue 93h
[page]
1993
Mathematical Reviews
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Metarecur- sion with an effective oracle, while having all of the advantages of Kreisel-Sacks metarecursion, also generalizes it to arbitrary pro- jectible countable ordinals.” ...
Logiki (1987), 145-156, 166; MR 90a:03070], the author introduces the notions of effective oracle and oracle com- putation in metarecursion theory [G. Kreisel and G. E. Sacks, J. ...
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