Intrinsic bounds on complexity and definability at limit levels

2009 ◽  
Vol 74 (3) ◽  
pp. 1047-1060 ◽  
Author(s):  
John Chisholm ◽  
Ekaterina B. Fokina ◽  
Sergey S. Goncharov ◽  
Valentina S. Harizanov ◽  
Julia F. Knight ◽  
...  
Keyword(s):  
Computable Structure ◽  
Additional Relation ◽  
Scott Family ◽  
Limit Ordinal

AbstractWe show that for every computable limit ordinal α, there is a computable structure that is categorical, but not relatively categorical (equivalently, it does not have a formally Scott family). We also show that for every computable limit ordinal α, there is a computable structure with an additional relation R that is intrinsically on , but not relatively intrinsically on (equivalently, it is not definable by a computable Σα formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α.

Scott Family: The Parentage Of Archbishop Rotherham

1877 ◽  
Vol s5-VII (182) ◽  
pp. 490-491
Author(s):  
James Greenstreet
Keyword(s):  
Scott Family

α-degrees of α-theories

1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
Author(s):  
George Metakides
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
The Other ◽  
Infinite Length ◽  
Infinitary Logic ◽  
Admissible Set ◽  
Recursively Enumerable ◽  
Limit Ordinal ◽  
The Subject ◽  
Further Development

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

Scott Family Enterprises (A): Defining Fair Process for Cousin Owners

2017 ◽  
pp. 1-24
Author(s):  
John L. Ward ◽  
Canh Tran
Keyword(s):  
Family Business ◽  
Large Family ◽  
Board Members ◽  
Fair Process ◽  
Family Expectations ◽  
Family Enterprises ◽  
Scott Family ◽  
Growing Pains ◽  
Generation Family ◽  
Made In

A large family business in banking and ranching is shifting leadership to the next generation and has developed a protocol to select board members by consensus. However, when the selection occurs, it is not made in accordance with the protocol, and a third-generation family member questions why the selection rules were changed by second-generation members without input or vote. Highlights the growing pains of developing fair processes and guidelines for nominating and selecting board members, meeting family expectations, communicating with constituents, and encouraging active roles in governance at the cousin-stage of a family business.

Scott family: the parentage of archbishop Rotherham

1878 ◽  
Vol s5-IX (229) ◽  
pp. 391-391
Author(s):  
W. Rotherham
Keyword(s):  
Scott Family

Gladstone and Scott: family, identity and nation

2007 ◽  
Vol 86 (1) ◽  
pp. 69-95
Author(s):  
Ruth. Clayton Windscheffel
Keyword(s):  
Family Identity ◽  
Scott Family

scholarly journals A Generalization of Final Rank of Primary Abelian Groups

1970 ◽  
Vol 22 (6) ◽  
pp. 1118-1122 ◽  
Author(s):  
Doyle O. Cutler ◽  
Paul F. Dubois
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Basic Subgroup ◽  
Primary Abelian Group ◽  
Limit Ordinal ◽  
Final Rank ◽  
Definition Of ◽  
Group Recall

Let G be a p-primary Abelian group. Recall that the final rank of G is infn∈ω{r(pnG)}, where r(pnG) is the rank of pnG and ω is the first limit ordinal. Alternately, if Γ is the set of all basic subgroups of G, we may define the final rank of G by supB∈Γ {r(G/B)}. In fact, it is known that there exists a basic subgroup B of G such that r(G/B) is equal to the final rank of G. Since the final rank of G is equal to the final rank of a high subgroup of G plus the rank of pωG, one could obtain the same information if the definition of final rank were restricted to the class of p-primary Abelian groups of length ω.

Scott family: The parentage of archbishop rotherham

1877 ◽  
Vol s5-VIII (204) ◽  
pp. 411-411 ◽  
Author(s):  
Edward Scott
Keyword(s):  
Scott Family

scholarly journals Enumerations in computable structure theory

2005 ◽  
Vol 136 (3) ◽  
pp. 219-246 ◽  
Author(s):  
Sergey Goncharov ◽  
Valentina Harizanov ◽  
Julia Knight ◽  
Charles McCoy ◽  
Russell Miller ◽  
...  
Keyword(s):  
Structure Theory ◽  
Computable Structure ◽  
Computable Structure Theory
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