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Metarecursively enumerable sets and admissible ordinals

Gerald E. Sacks
1966 Bulletin of the American Mathematical Society  
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

Gerald E. Sacks
1966 Transactions of the American Mathematical Society  
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  
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

Gerald E Sacks
1971 Advances in Mathematics  
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  
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  
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  
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

Noam Greenberg
2005 Bulletin of Symbolic Logic  
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

Noam Greenberg
2006 Memoirs of the American Mathematical Society  
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  
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 $\mathcal{O}$

Harvey Friedman
1976 Proceedings of the American Mathematical Society  
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  
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)  
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)  
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  
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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