scholarly journals Lovely pairs of models: the non first order case

2004 ◽  
Vol 69 (3) ◽  
pp. 641-662 ◽  
Author(s):  
Itay Ben-Yaacov
Keyword(s):  
Order Theory ◽  
Simple Theory ◽  
Abstract Theory ◽  
First Order ◽  
Saturated Models ◽  
First Order Theory ◽  
Order Case

Abstract.We prove that for every simple theory T (or even simple thick compact abstract theory) there is a (unique) compact abstract theory whose saturated models are the lovely pairs of T. Independence-theoretic results that were proved in [5] when is a first order theory are proved for the general case: in particular is simple and we characterise independence.

scholarly journals Categoricity and U-rank in excellent classes

2003 ◽  
Vol 68 (4) ◽  
pp. 1317-1336 ◽  
Author(s):  
Olivier Lessmann
Keyword(s):  
Free Groups ◽  
Order Theory ◽  
First Order ◽  
Atomic Models ◽  
First Order Theory ◽  
Uncountable Cardinal ◽  
Order Case ◽  
Uncountable Cardinals

AbstractLet be the class of atomic models of a countable first order theory. We prove that if is excellent and categorical in some uncountable cardinal, then each model is prime and minimal over the basis of a definable pregeometry given by a quasiminimal set. This implies that is categorical in all uncountable cardinals. We also introduce a U-rank to measure the complexity of complete types over models. We prove that the U-rank has the usual additivity properties, that quasiminimal types have U-rank 1, and that the U-rank of any type is finite in the uncountably categorical, excellent case. However, in contrast to the first order case, the supremum of the U-rank over all types may be ω (and is not achieved). We illustrate the theory with the example of free groups, and Zilber's pseudo analytic structures.

A GEOMETRIC INTRODUCTION TO FORKING AND THORN-FORKING

2009 ◽  
Vol 09 (01) ◽  
pp. 1-20 ◽  
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.

The consistency of some 4-stratified subsystem of NF including NF3

1985 ◽  
Vol 50 (2) ◽  
pp. 407-411 ◽  
Author(s):  
Maurice Boffa ◽  
Paolo Casalegno
Keyword(s):  
Order Theory ◽  
Simple Theory ◽  
Natural Numbers ◽  
First Order ◽  
First Order Theory

As is well known, NF is a first-order theory whose language coincides with that of ZF. The nonlogical axioms of the theory are: Extensionality. (x)(y)[(z)(z ∈ x ↔ z ∈ y) → x = y].Comprehension. (Ex)(y)(y ∈ x ↔ ψ) for every stratified ψ in which x does not occur free (a formula of NF is said to be stratified if it can be turned into a formula of the simple theory of types by adding type indices (natural numbers ≥ 0) to its variables).Before stating our result, a few preliminaries are in order. Let T be the simple theory of types. If ψ is a formula of T, we denote by ψ+ the formula obtained from ψ by raising all type indices by 1. T* is the result of adding to T every axiom of the form ψ ↔ ψ+. A formula of T is n-stratified (n > 0) if it does not contain any type index ≥ n. A formula of NF is n-stratified if it can be turned into an n-stratified formula of T by adding type indices to its variables. (In practice, we shall allow ourselves to confuse an n-stratified formula of T with the corresponding n-stratified formula of NF). For n > 0, Tn (resp. ) is the subtheory of T (resp. T*) containing only n-stratified formulae. For n > 0, NFn is the subtheory of NF generated by those axioms of NF which are n-stratified. Let = 〈M0, M1,…,=, ∈〉 be a model of T.

POSITIVE MODEL THEORY AND COMPACT ABSTRACT THEORIES

2003 ◽  
Vol 03 (01) ◽  
pp. 85-118 ◽  
Author(s):  
ITAY BEN-YAACOV
Keyword(s):  
Model Theory ◽  
Hilbert Spaces ◽  
Classical Model ◽  
Order Theory ◽  
Abstract Theory ◽  
First Order ◽  
First Order Theory

We develop positive model theory, which is a non first order analogue of classical model theory where compactness is kept at the expense of negation. The analogue of a first order theory in this framework is a compact abstract theory: several equivalent yet conceptually different presentations of this notion are given. We prove in particular that Banach and Hilbert spaces are compact abstract theories, and in fact very well-behaved as such.

scholarly journals A first-order theory of Ulm type

Computability ◽  
2019 ◽  
Vol 8 (3-4) ◽  
pp. 347-358
Author(s):  
Matthew Harrison-Trainor
Keyword(s):  
Order Theory ◽  
First Order ◽  
First Order Theory

scholarly journals Quasi-decidability of a Fragment of the First-Order Theory of Real Numbers

2015 ◽  
Vol 57 (2) ◽  
pp. 157-185 ◽  
Author(s):  
Peter Franek ◽  
Stefan Ratschan ◽  
Piotr Zgliczynski
Keyword(s):  
Order Theory ◽  
Real Numbers ◽  
First Order ◽  
First Order Theory
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