Definable types in -minimal theories

1994 ◽  
Vol 59 (1) ◽  
pp. 185-198 ◽  
Author(s):  
David Marker ◽  
Charles I. Steinhorn
Keyword(s):  
Stability Theory ◽  
Conservative Extension ◽  
Absolute Value ◽  
First Order ◽  
Real Closed Fields ◽  
Order Language ◽  
Closed Fields ◽  
Models Of Arithmetic

Let L be a first order language. If M is an L-structure, let LM be the expansion of L obtained by adding constants for the elements of M.Definition. A type is definable if and only if for any L-formula , there is an LM-formula so that for all iff M ⊨ dθ(¯). The formula dθ is called the definition of θ.Definable types play a central role in stability theory and have also proven useful in the study of models of arithmetic. We also remark that it is well known and easy to see that for M ≺ N, the property that every M-type realized in N is definable is equivalent to N being a conservative extension of M, whereDefinition. If M ≺ N, we say that N is a conservative extension of M if for any n and any LN -definable S ⊂ Nn, S ∩ Mn is LM-definable in M.Van den Dries [Dl] studied definable types over real closed fields and proved the following result.0.1 (van den Dries), (i) Every type over (R, +, -,0,1) is definable.(ii) Let F and K be real closed fields and F ⊂ K. Then, the following are equivalent:(a) Every element of K that is bounded in absolute value by an element of F is infinitely close (in the sense of F) to an element of F.(b) K is a conservative extension of F.

scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

Omitting types in -minimal theories

1986 ◽  
Vol 51 (1) ◽  
pp. 63-74 ◽  
Author(s):  
David Marker
Keyword(s):  
Finite Union ◽  
Structure Theory ◽  
Real Closed Fields ◽  
Minimal Theory ◽  
Order Language ◽  
Closed Fields ◽  
Hanf Number ◽  
Definable Sets ◽  
Omitting Types ◽  
Binary Relation Symbol

Let L be a first order language containing a binary relation symbol <.Definition. Suppose ℳ is an L-structure and < is a total ordering of the domain of ℳ. ℳ is ordered minimal (-minimal) if and only if any parametrically definable X ⊆ ℳ can be represented as a finite union of points and intervals with endpoints in ℳ.In any ordered structure every finite union of points and intervals is definable. Thus the -minimal structures are the ones with no unnecessary definable sets. If T is a complete L-theory we say that T is strongly (-minimal if and only if every model of T is -minimal.The theory of real closed fields is the canonical example of a strongly -minimal theory. Strongly -minimal theories were introduced (in a less general guise which we discuss in §6) by van den Dries in [1]. Extending van den Dries' work, Pillay and Steinhorn (see [3], [4] and [2]) developed an extensive structure theory for definable sets in strongly -minimal theories, generalizing the results for real closed fields. They also established several striking analogies between strongly -minimal theories and ω-stable theories (most notably the existence and uniqueness of prime models). In this paper we will examine the construction of models of strongly -minimal theories emphasizing the problems involved in realizing and omitting types. Among other things we will prove that the Hanf number for omitting types for a strongly -minimal theory T is at most (2∣T∣)+, and characterize the strongly -minimal theories with models order isomorphic to (R, <).

From Stability to Simplicity

1998 ◽  
Vol 4 (1) ◽  
pp. 17-36 ◽  
Author(s):  
Byunghan Kim ◽  
Anand Pillay
Keyword(s):  
Recent Work ◽  
Stability Theory ◽  
Simple Theories ◽  
First Order ◽  
Closed Fields ◽  
Exhaustive Study ◽  
Pseudofinite Fields ◽  
Algebraically Closed Fields ◽  
Major Progress ◽  
Made In

§1. Introduction. In this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the simple theories. Major progress was made in the first author's doctoral thesis [17]. We will give a survey of this, as well as further works by the authors and others.The class of simple theories includes stable theories, but also many more, such as the theory of the random graph. Moreover, many of the theories of particular algebraic structures which have been studied recently (pseudofinite fields, algebraically closed fields with a generic automorphism, smoothly approximable structures) turn out to be simple. The interest is basically that a large amount of the machinery of stability theory, invented by Shelah, is valid in the broader class of simple theories. Stable theories will be defined formally in the next section. An exhaustive study of them is carried out in [33]. Without trying to read Shelah's mind, we feel comfortable in saying that the importance of stability for Shelah lay partly in the fact that an unstable theory T has 2λ many models in any cardinal λ ≥ ω1 + |T| (proved by Shelah). (Note that for λ ≥ |T| 2λ is the maximum possible number of models of cardinality λ.)

Corrigendum to: “Real closed fields and models of arithmetic”

2012 ◽  
Vol 77 (2) ◽  
pp. 726-726 ◽  
Author(s):  
P. D'Aquino ◽  
J. F. Knight ◽  
S. Starchenko
Keyword(s):  
Real Closed Fields ◽  
Closed Fields ◽  
Models Of Arithmetic

Quantifier elimination in separably closed fields of finite imperfectness degree

1988 ◽  
Vol 53 (2) ◽  
pp. 463-469 ◽  
Author(s):  
Dan Haran
Keyword(s):  
Quantifier Elimination ◽  
Extension Property ◽  
Recursive Procedure ◽  
Prime Extension ◽  
First Order ◽  
Linearly Independent ◽  
Order Language ◽  
Closed Fields ◽  
Primitive Recursive

The theory of separably closed fields of a fixed characteristic and a fixed imperfectness degree is clearly recursively axiomatizable. Ershov [1] showed that it is complete, and therefore decidable. Later it became clear that this theory also has the prime extension property in a suitable language (cf. [4, Proposition 1]); hence it admits quantifier elimination. The purpose of this work is to give an explicit, primitive recursive procedure for such quantifier elimination in the case of a finite imperfectness degree.To be precise, the language ∧ that we have in mind is the first order language of fields enriched with (m + 1)-place function symbols , where m = 0,1,2,… and 1 ≤ j ≤ pm. To interpret in a field M of characteristic p, consider the p-adic expansion of j – 1, and for x1,…,xm Є M let . If x1, …, xm) are p-independent and y Є M is p-dependent on them, then are linearly independent over Mp and y is linearly dependent on them. In this case there are unique such that define . Set otherwise.Denote by SCF(p,e) the theory of separably closed fields of characteristic p and finite imperfectness degree e, containing the above interpretation of the functions .

scholarly journals The Role of True Finiteness in the Admissible Recursively Enumerable Degrees

2005 ◽  
Vol 11 (3) ◽  
pp. 398-410
Author(s):  
Noam Greenberg
Keyword(s):  
Partially Ordered Sets ◽  
Recursion Theory ◽  
Ordered Sets ◽  
First Order ◽  
Recursively Enumerable ◽  
Order Language ◽  
Replacement Scheme ◽  
Admissible Ordinals ◽  
Models Of Arithmetic

AbstractWhen attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of α-finiteness. As examples we discuss both codings of models of arithmetic into the recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. We show that if an admissible ordinal α is effectively close to ω (where this closeness can be measured by size or by cofinality) then such constructions may be performed in the α-r.e. degrees, but otherwise they fail. 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 of α-r.e. degrees for various classes of admissible ordinals α. Together with coding work which shows that for some α, the theory of the α-r.e. degrees is complicated, we get that for every admissible ordinal α, the α-r.e. degrees and the classical r.e. degrees are not elementarily equivalent.

scholarly journals On the elementary theory of pairs of real closed fields. II

1982 ◽  
Vol 47 (3) ◽  
pp. 669-679 ◽  
Author(s):  
Walter Baur
Keyword(s):  
Power Series ◽  
Abelian Groups ◽  
Elementary Theory ◽  
Predicate Symbol ◽  
Axiom System ◽  
Closed Field ◽  
Algebraic Numbers ◽  
Real Closed Fields ◽  
Order Language ◽  
Closed Fields

Let ℒ be the first order language of field theory with an additional one place predicate symbol. In [B2] it was shown that the elementary theory T of the class of all pairs of real closed fields, i.e., ℒ-structures ‹K, L›, K a real closed field, L a real closed subfield of K, is undecidable.The aim of this paper is to show that the elementary theory Ts of a nontrivial subclass of containing many naturally occurring pairs of real closed fields is decidable (Theorem 3, §5). This result was announced in [B2]. An explicit axiom system for Ts will be given later. At this point let us just mention that any model of Ts, is elementarily equivalent to a pair of power series fields ‹R0((TA)), R1((TB))› where R0 is the field of real numbers, R1 = R0 or the field of real algebraic numbers, and B ⊆ A are ordered divisible abelian groups. Conversely, all these pairs of power series fields are models of Ts.Theorem 3 together with the undecidability result in [B2] answers some of the questions asked in Macintyre [M]. The proof of Theorem 3 uses the model theoretic techniques for valued fields introduced by Ax and Kochen [A-K] and Ershov [E] (see also [C-K]). The two main ingredients are(i) the completeness of the elementary theory of real closed fields with a distinguished dense proper real closed subfield (due to Robinson [R]),(ii) the decidability of the elementary theory of pairs of ordered divisible abelian groups (proved in §§1-4).I would like to thank Angus Macintyre for fruitful discussions concerning the subject. The valuation theoretic method of classifying theories of pairs of real closed fields is taken from [M].

Définissabilité dans les corps de fonctions p-adiques

1991 ◽  
Vol 56 (3) ◽  
pp. 783-785 ◽  
Author(s):  
Luc Bélair ◽  
Jean-Louis Duret
Keyword(s):  
Function Fields ◽  
Elementary Property ◽  
First Order ◽  
Order Language ◽  
Closed Fields

AbstractWe study function fields over p-adically closed fields in the first-order language of fields. Using ideas of Duret [D], we show that the field of constants is definable, and that the genus is an elementary property.

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