THORN-FORKING AS LOCAL FORKING

We introduce the notion of a preindependence relation between subsets of the big model of a complete first-order theory, an abstraction of the properties which numerous concrete notions such as forking, dividing, thorn-forking, thorn-dividing, splitting or finite satisfiability share in all complete theories. We examine the relation between four additional axioms (extension, local character, full existence and symmetry) that one expects of a good notion of independence. We show that thorn-forking can be described in terms of local forking if we localize the number k in Kim's notion of "dividing with respect to k" (using Ben-Yaacov's "k-inconsistency witnesses") rather than the forking formulas. It follows that every theory with an M-symmetric lattice of algebraically closed sets (in Teq) is rosy, with a simple lattice theoretical interpretation of thorn-forking.

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A GEOMETRIC INTRODUCTION TO FORKING AND THORN-FORKING

Journal of Mathematical Logic ◽

10.1142/s0219061309000811 ◽

2009 ◽

Vol 09 (01) ◽

pp. 1-20 ◽

Cited By ~ 23

Author(s):

HANS ADLER

Keyword(s):

Order Theory ◽

Simple Theory ◽

Ternary Relation ◽

Mathematical Objects ◽

Main Application ◽

First Order ◽

Closed Sets ◽

First Order Theory ◽

Independence Relations

A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be closely related to modular pairs in the lattice of algebraically closed sets.

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Coding true arithmetic in the Medvedev and Muchnik degrees

Journal of Symbolic Logic ◽

10.2178/jsl/1294171000 ◽

2011 ◽

Vol 76 (1) ◽

pp. 267-288 ◽

Cited By ~ 5

Author(s):

Paul Shafer

Keyword(s):

Order Theory ◽

Second Order ◽

Third Order ◽

First Order ◽

Closed Sets ◽

The Third ◽

First Order Theory ◽

Medvedev Degrees

AbstractWe prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order theory of true arithmetic are pairwise recursively isomorphic (obtained independently by Lewis, Nies, and Sorbi [7]). We then restrict our attention to the degrees of closed sets and prove that the following theories are pairwise recursively isomorphic: the first-order theory of the closed Medvedev degrees, the first-order theory of the compact Medvedev degrees, the first-order theory of the closed Muchnik degrees, the first-order theory of the compact Muchnik degrees, and the second-order theory of true arithmetic. Our coding methods also prove that neither the closed Medvedev degrees nor the compact Medvedev degrees are elementarily equivalent to either the closed Muchnik degrees or the compact Muchnik degrees.

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Complete theories with only universal and existential axioms

Journal of Symbolic Logic ◽

10.1017/s0022481200029704 ◽

1987 ◽

Vol 52 (3) ◽

pp. 698-711 ◽

Cited By ~ 1

Author(s):

A. H. Lachlan

Keyword(s):

Order Theory ◽

First Order ◽

First Order Theory ◽

Existential Sentences ◽

Complete Theories ◽

The Universe

AbstractLet T be a complete first-order theory over a finite relational language which is axiomatized by universal and existential sentences. It is shown that T is almost trivial in the sense that the universe of any model of T can be written . where F is finite and I1, I2, …, In are mutually indiscernible over F. Some results about complete theories with ∃∀-axioms over a finite relational language are also mentioned.

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