An autostable 1-decidable model without a computable Scott family of ∃-formulas

O. V. Kudinov

doi:10.1007/bf02367027

Scott Family: The Parentage Of Archbishop Rotherham

Notes and Queries ◽

10.1093/nq/s5-vii.182.490a ◽

1877 ◽

Vol s5-VII (182) ◽

pp. 490-491

Author(s):

James Greenstreet

Keyword(s):

Scott Family

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Decidable Model Companions

Mathematical Logic Quarterly ◽

10.1002/malq.19890350305 ◽

1989 ◽

Vol 35 (3) ◽

pp. 225-227 ◽

Cited By ~ 3

Author(s):

Stanley Burris

Keyword(s):

Decidable Model

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Indiscernibles and decidable models

Journal of Symbolic Logic ◽

10.2307/2273316 ◽

1983 ◽

Vol 48 (1) ◽

pp. 21-32 ◽

Cited By ~ 6

Author(s):

H. A. Kierstead ◽

J. B. Remmel

Keyword(s):

Order Theory ◽

Stable Theory ◽

Categorical Theory ◽

Large Classes ◽

Decidable Theory ◽

First Order Theory ◽

Decidable Model ◽

Decidable Theories ◽

Basic Facts ◽

Infinite Set

Ehrenfeucht and Mostowski [3] introduced the notion of indiscernibles and proved that every first order theory has a model with an infinite set of order indiscernibles. Since their work, techniques involving indiscernibles have proved to be extremely useful for constructing models with various specialized properties. In this paper and in a sequel [5], we investigate the effective content of Ehrenfeucht's and Mostowski's result. In this paper we consider the question of which decidable theories have decidable models with infinite recursive sets of indiscernibles. In §1, using some basic facts from stability theory, we show that certain large classes of decidable theories have decidable models with infinite recursive sets of indiscernibles. For example, we show that every ω-stable decidable theory and every stable theory which possesses a certain strong decidability property called ∃Q-decidability have such models. In §2 we construct several examples of decidable theories which have no decidable models with infinite recursive sets of indiscernibles. These examples show that our hypothesis for our positive results in §1 are necessary. Finally in §3 we give two applications of our results. First as an easy application of our results in §1, we show that every ω-stable decidable theory has uncountable models which realize only recursive types. Also our counterexamples in §2 allow us to answer negatively two questions of Baldwin and Kueker [1] concerning the effectiveness of their elimination of Ramsey quantifiers for certain theories.In [5], we show that in general the problem of finding an infinite set of indiscernibles in a decidable model is recursively equivalent to finding a path through a recursive infinite branching tree. Similarly, we show that the problem of finding an co-type of a set of indiscernibles in a decidable ω-categorical theory is recursively equivalent to finding a path through a highly recursive finitely branching tree.

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

Kellogg School of Management Cases ◽

10.1108/case.kellogg.2016.000296 ◽

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.

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