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Metarecursively enumerable sets and admissible ordinals

1966
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Bulletin of the American Mathematical Society
*

It was observed in [8] that there exist bounded,

doi:10.1090/s0002-9904-1966-11416-7
fatcat:howe4vmtibf45dtvilxup5ktce
*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. ...##
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Post's problem, admissible ordinals, and regularity

1966
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Transactions of the American Mathematical Society
*

Theorem 4 of [7] states that there exist two

doi:10.1090/s0002-9947-1966-0201299-1
fatcat:etht5unbhvajnitfuehznm6aia
*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. ...##
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Page 742 of Mathematical Reviews Vol. 35, Issue 4
[page]

1968
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Mathematical Reviews
*

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 ...##
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On the reducibility of ⊓11 sets

1971
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Advances in Mathematics
*

A

doi:10.1016/0001-8708(71)90042-9
fatcat:rmf5tkwiy5ewxnfevbwlbl7dxu
*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. ...##
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Page 16 of Mathematical Reviews Vol. 44, Issue 1
[page]

1972
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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. ...##
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Page 1289 of Mathematical Reviews Vol. 40, Issue 6
[page]

1970
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Mathematical Reviews
*

*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*. ...

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Page 11 of Mathematical Reviews Vol. 43, Issue 1
[page]

1972
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Mathematical Reviews
*

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 ...##
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The Role of True Finiteness in the Admissible Recursively Enumerable Degrees

2005
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Bulletin of Symbolic Logic
*

The results of these constructions can be expressed in the first-order language of partially ordered

doi:10.2178/bsl/1122038994
fatcat:5h7jio3leza53nvscnmymv2yum
*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. ...##
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The role of true finiteness in the admissible recursively enumerable degrees

2006
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Memoirs of the American Mathematical Society
*

The results of these constructions can be expressed in the first-order language of partially ordered

doi:10.1090/memo/0854
fatcat:kxkbxy6jwbavxo5asuffdfgnom
*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. ...##
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Page 44 of Mathematical Reviews Vol. , Issue 90A
[page]

1990
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Mathematical Reviews
*

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. ...##
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Recursiveness in $P^1_1$ paths through $\mathcal{O}$

1976
*
Proceedings of the American Mathematical Society
*

If every hyp

doi:10.1090/s0002-9939-1976-0398812-2
fatcat:cw4d3xwmkjhxbop2fhpveyfn3q
*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. ...##
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Page 35 of Mathematical Reviews Vol. , Issue 92a
[page]

1992
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Mathematical Reviews
*

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 ...##
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JSL volume 32 issue 1 Cover and Front matter

1967
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Journal of Symbolic Logic (JSL)
*

By MARIKO YASUGI 145 Simplicity of recursively enumerable

doi:10.1017/s0022481200114501
fatcat:zezl2axhirfjlbi2xi5fhanqb4
*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. ...##
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Abstracts of papers

1966
*
Journal of Symbolic Logic (JSL)
*

Some applications of

doi:10.1017/s0022481200126258
fatcat:y5dyvpw3cffcxikdbyhmmciyou
*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). ...##
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Page 4135 of Mathematical Reviews Vol. , Issue 93h
[page]

1993
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Mathematical Reviews
*

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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