T-convexity and tame extensions II

Lou van den Dries

doi:10.2307/2275729

T-convexity and tame extensions II

Journal of Symbolic Logic ◽

10.2307/2275729 ◽

1997 ◽

Vol 62 (1) ◽

pp. 14-34 ◽

Cited By ~ 10

Author(s):

Lou van den Dries

Keyword(s):

Residue Class ◽

Residue Field ◽

Elementary Substructure ◽

Real Closed Fields ◽

Minimal Theory ◽

Ordered Field ◽

Closed Fields

I solve here some problems left open in “T-convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular, T will denote a complete o-minimal theory extending RCF, the theory of real closed fields. Let (, V) ⊨ Tconvex, let = V/m(V) be the residue field, with residue class map x ↦ : V ↦ , and let υ: → Γ be the associated valuation. “Definable” will mean “definable with parameters”. The main goal of this article is to determine the structure induced by (, V) on its residue fieldand on its value group Γ. In [9] we expanded the ordered field to a model of T as follows. Take a tame elementary substructure ′ of such that R′ ⊆ V and R′ maps bijectively onto under the residue class map, and make this bijection into an isomorphism ′ ≌ of T-models. (We showed such ′ exists, and that this gives an expansion of to a T-model that is independent of the choice of ′.).

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The domination monoid in o-minimal theories

Journal of Mathematical Logic ◽

10.1142/s0219061321500306 ◽

2021 ◽

Author(s):

Rosario Mennuni

Keyword(s):

Residue Field ◽

Formal Logic ◽

Valued Fields ◽

Real Closed Fields ◽

Minimal Theory ◽

Closed Fields ◽

Text Types ◽

Global Invariant

We study the monoid of global invariant types modulo domination-equivalence in the context of o-minimal theories. We reduce its computation to the problem of proving that it is generated by classes of [Formula: see text]-types. We show this to hold in Real Closed Fields, where generators of this monoid correspond to invariant convex subrings of the monster model. Combined with [C. Ealy, D. Haskell and J. Maríková, Residue field domination in real closed valued fields, Notre Dame J. Formal Logic 60(3) (2019) 333–351], this allows us to compute the domination monoid in the weakly o-minimal theory of Real Closed Valued Fields.

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T-convexity and tame extensions

Journal of Symbolic Logic ◽

10.2307/2275510 ◽

1995 ◽

Vol 60 (1) ◽

pp. 74-102 ◽

Cited By ~ 13

Author(s):

Lou van den Dries ◽

Adam H. Lewenberg

Keyword(s):

Convex Hull ◽

Residue Field ◽

Quantifier Elimination ◽

Convex Hulls ◽

Minimal Extension ◽

Elementary Substructure ◽

Real Closed Fields ◽

Closed Fields ◽

Natural Expansion

AbstractLet T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that the residue field of such a convex hull has a natural expansion to a model of T. We give a quantifier elimination relative to T for the theory of pairs (ℛ, V) where ℛ ⊨ T and V ≠ ℛ is the convex hull of an elementary substructure of ℛ. We deduce that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to T for the theory of pairs with ℛ a model of T and a proper elementary substructure that is Dedekind complete in ℛ. We deduce that the theory of such “tame” pairs is complete.

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Omitting types in -minimal theories

Journal of Symbolic Logic ◽

10.2307/2273943 ◽

1986 ◽

Vol 51 (1) ◽

pp. 63-74 ◽

Cited By ~ 10

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, <).

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Relative randomness and real closed fields

Journal of Symbolic Logic ◽

10.2178/jsl/1107298522 ◽

2005 ◽

Vol 70 (1) ◽

pp. 319-330 ◽

Cited By ~ 5

Author(s):

Alexander Raichev

Keyword(s):

Real Number ◽

Real Closed Field ◽

Closed Field ◽

Real Numbers ◽

The Real ◽

Real Closed Fields ◽

Ordered Field ◽

Closed Fields ◽

Master's Thesis ◽

National University

AbstractWe show that for any real number, the class of real numbers less random than it, in the sense of rK-reducibility, forms a countable real closed subfield of the real ordered field. This generalizes the well-known fact that the computable reals form a real closed field.With the same technique we show that the class of differences of computably enumerable reals (d.c.e. reals) and the class of computably approximable reals (c.a. reals) form real closed fields. The d.c.e. result was also proved nearly simultaneously and independently by Ng (Keng Meng Ng, Master's Thesis, National University of Singapore, in preparation).Lastly, we show that the class of d.c.e. reals is properly contained in the class or reals less random than Ω (the halting probability), which in turn is properly contained in the class of c.a. reals, and that neither the first nor last class is a randomness class (as captured by rK-reducibility).

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A note on valuation definable expansions of fields

Journal of Symbolic Logic ◽

10.2307/2586860 ◽

1998 ◽

Vol 63 (2) ◽

pp. 739-743 ◽

Cited By ~ 1

Author(s):

Deirdre Haskell ◽

Dugald Macpherson

Keyword(s):

Valuation Ring ◽

Quantifier Elimination ◽

Real Closed Field ◽

Closed Field ◽

Real Closed Fields ◽

Ordered Field ◽

Valued Field ◽

Closed Fields ◽

Binary Predicate ◽

Algebraically Closed Fields

In this note, we consider models of the theories of valued algebraically closed fields and convexly valued real closed fields, their reducts to the pure field or ordered field language respectively, and expansions of these by predicates which are definable in the valued field. We show that, in terms of definability, there is no structure properly between the pure (ordered) field and the valued field. Our results are analogous to several other definability results for reducts of algebraically closed and real closed fields; see [9], [10], [11] and [12]. Throughout this paper, definable will mean definable with parameters.Theorem A. Let ℱ = (F, +, ×, V) be a valued, algebraically closed field, where V denotes the valuation ring. Let A be a subset ofFndefinable in ℱv. Then either A is definable in ℱ = (F, +, ×) or V is definable in.Theorem B. Let ℛv = (R, <, +, ×, V) be a convexly valued real closed field, where V denotes the valuation ring. Let Abe a subset ofRndefinable in ℛv. Then either A is definable in ℛ = (R, <, +, ×) or V is definable in.The proofs of Theorems A and B are quite similar. Both ℱv and ℛv admit quantifier elimination if we adjoin a definable binary predicate Div (interpreted by Div(x, y) if and only if v(x) ≤ v(y)). This is proved in [14] (extending [13]) in the algebraically closed case, and in [4] in the real closed case. We show by direct combinatorial arguments that if the valuation is not definable then the expanded structure is strongly minimal or o-minimal respectively. Then we call on known results about strongly minimal and o-minimal fields to show that the expansion is not proper.

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Real closed fields and models of Peano arithmetic

Journal of Symbolic Logic ◽

10.2178/jsl/1264433906 ◽

2012 ◽

Vol 75 (1) ◽

pp. 1-11 ◽

Cited By ~ 4

Author(s):

P. D'Aquino ◽

J. F. Knight ◽

S. Starchenko

Keyword(s):

Peano Arithmetic ◽

Integer Part ◽

The Real ◽

Real Closed Fields ◽

Nonstandard Model ◽

Ordered Field ◽

Closed Fields ◽

Real Closure ◽

Ordered Ring ◽

Recursively Saturated

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of I Open iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ4), then R must be recursively saturated. In particular, the real closure of I, RC (I), is recursively saturated. We also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA.

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Vaught's conjecture for o-minimal theories

Journal of Symbolic Logic ◽

10.1017/s0022481200028991 ◽

1988 ◽

Vol 53 (1) ◽

pp. 146-159 ◽

Cited By ~ 11

Author(s):

Laura L. Mayer

Keyword(s):

Structure Theorem ◽

Classification Theory ◽

Real Closed Fields ◽

Minimal Theory ◽

Definable Subset ◽

Closed Fields ◽

The Universe ◽

Countable Models

The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is a model of T then M is linearly ordered and every definable subset of the universe of M consists of finitely many intervals and points. The theory of real closed fields is an example of an o-minimal theory.We examine the structure of the countable models for T, T an arbitrary o-minimal theory (in a countable language). We completely characterize these models, provided that T does not have 2ω countable models. This proviso (viz. that T has fewer than 2ω countable models) is in the tradition of classification theory: given a cardinal α, if T has the maximum possible number of models of size α, i.e. 2α, then no structure theorem is expected (cf. [Sh1]).O-minimality is introduced in §1. §1 also contains conventions and definitions, including the definitions of cut and noncut. Cuts and noncuts constitute the nonisolated types over a set.In §2 we study a notion of independence for sets of nonisolated types and the corresponding notion of dimension.In §3 we define what it means for a nonisolated type to be simple. Such types generalize the so-called “components” in Pillay and Steinhorn's analysis of ω-categorical o-minimal theories [PS]. We show that if there is a nonisolated type which is not simple then T has 2ω countable models.

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