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.)

Ordinal spectra of first-order theories

1977 ◽  
Vol 42 (4) ◽  
pp. 492-505 ◽  
Author(s):  
John Stewart Schlipf
Keyword(s):  
Recent Work ◽  
Information Content ◽  
Recursion Theory ◽  
Admissible Set ◽  
First Order ◽  
Admissible Sets ◽  
Closed Fields ◽  
Algebraically Closed Fields ◽  
Basic Treatment

The notion of the next admissible set has proved to be a very useful notion in definability theory and generalized recursion theory, a unifying notion that has produced further interesting results in its own right. The basic treatment of the next admissible set above a structure ℳ of urelements is to be found in Barwise's [75] book Admissible sets and structures. Also to be found there are many of the interesting characterizations of the next admissible set. For further justification of the interest of the next admissible set the reader is referred to Moschovakis [74], Nadel and Stavi [76] and Schlipf [78a, b, c].One of the most interesting single properties of is its ordinal (ℳ). It coincides, for example, with Moschovakis' inductive closure ordinal over structures ℳ with pairing functions—and over some, such as algebraically closed fields of characteristic 0, without pairing functions (by recent work of Arthur Rubin) (although a locally famous counterexample of Kunen, a theorem of Barwise [77], and some recent results of Rubin and the author, show that the inductive closure ordinal may also be strictly smaller in suitably pathological structures). Further justification for looking at (ℳ) alone may be found in the above-listed references. Loosely, we can consider the size of to be a useful measure of the complexity of ℳ. One of the simplest measures of the size of —and yet a very useful measure—is its ordinal, (ℳ). Keisler has suggested thinking of (ℳ) as the information content of a model—the supremum of lengths of wellfounded relations characterizable in the model.

Differentially algebraic group chunks

1990 ◽  
Vol 55 (3) ◽  
pp. 1138-1142 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Algebraic Group ◽  
Characteristic Zero ◽  
Quantifier Elimination ◽  
Closed Field ◽  
Zariski Topology ◽  
First Order ◽  
Differential Field ◽  
Closed Fields ◽  
Differential Fields ◽  
Algebraically Closed Fields

We point out that a group first order definable in a differentially closed field K of characteristic 0 can be definably equipped with the structure of a differentially algebraic group over K. This is a translation into the framework of differentially closed fields of what is known for groups definable in algebraically closed fields (Weil's theorem).I restrict myself here to showing (Theorem 20) how one can find a large “differentially algebraic group chunk” inside a group defined in a differentially closed field. The rest of the translation (Theorem 21) follows routinely, as in [B].What is, perhaps, of interest is that the proof proceeds at a completely general (soft) model theoretic level, once Facts 1–4 below are known.Fact 1. The theory of differentially closed fields of characteristic 0 is complete and has quantifier elimination in the language of differential fields (+, ·,0,1, −1,d).Fact 2. Affine n-space over a differentially closed field is a Noetherian space when equipped with the differential Zariski topology.Fact 3. If K is a differentially closed field, k ⊆ K a differential field, and a and are in k, then a is in the definable closure of k ◡ iff a ∈ ‹› (where k ‹› denotes the differential field generated by k and).Fact 4. The theory of differentially closed fields of characteristic zero is totally transcendental (in particular, stable).

The model-companion of a class of structures

1972 ◽  
Vol 37 (3) ◽  
pp. 546-556 ◽  
Author(s):  
G. L. Cherlin
Keyword(s):  
Order Theory ◽  
Model Companion ◽  
Elementary Substructure ◽  
Closed Model ◽  
First Order ◽  
First Order Theory ◽  
Logical Equivalence ◽  
Closed Fields ◽  
Algebraically Closed Fields ◽  
Model Completion

If Σ is the class of all fields and Σ* is the class of all algebraically closed fields, then it is well known that Σ* is characterized by the following properties:(i) Σ* is the class of models of some first order theory K*.(ii) If m1m2 are in Σ* and m1 ⊆ m2 then m1 ≺ m2 (m1 is an elementary substructure of m2, i.e. any first order sentence true in m1 is true in m2).(iii) If m1 is in Σ then there is a structure m2 in Σ* such that m1 ⊆ m2.If Σ is some other class of models of a first order theory K and a subclass Σ* of Σ exists satisfying (i)–(iii) then Σ* is uniquely determined and K* (which is unique up to logical equivalence) is called the model-companion of K. This notion is a generalization of the fundamental notion of model-completion introduced and extensively studied by A. Robinson [6], When the model-companion exists it provides the basis for a satisfactory treatment of the notion of an algebraically closed model of K.Recently A. Robinson has developed a more general formulation of the notion of “algebraically closed” structures in Σ, which is applicable to any inductive elementary class Σ of structures (by elementary we always mean ECΔ). Condition (i) must be weakened to(i′) Σ* is closed under elementary substructure (i.e. if m1 is in Σ* and m2 ≺ m1 then m2 is in Σ*).

Properties of forking in ω-free pseudo-algebraically closed fields

2002 ◽  
Vol 67 (3) ◽  
pp. 957-996 ◽  
Author(s):  
Zoé Chatzidakis
Keyword(s):  
Galois Group ◽  
Finite Fields ◽  
Order Theory ◽  
Absolute Galois Group ◽  
Prime Field ◽  
First Order ◽  
First Order Theory ◽  
Closed Fields ◽  
The Absolute ◽  
Algebraically Closed Fields

The study of pseudo-algebraically closed fields (henceforth called PAC) started with the work of J. Ax on finite and pseudo-finite fields [1]. He showed that the infinite models of the theory of finite fields are exactly the perfect PAC fields with absolute Galois group isomorphic to , and gave elementary invariants for their first order theory, thereby proving the decidability of the theory of finite fields. Ax's results were then extended to a larger class of PAC fields by M. Jarden and U. Kiehne [21], and Jarden [19]. The final word on theories of PAC fields was given by G. Cherlin, L. van den Dries and A. Macintyre [10], see also results by Ju. Ershov [13], [14]. Let K be a PAC field. Then the elementary theory of K is entirely determined by the following data:• The isomorphism type of the field of absolute numbers of K (the subfield of K of elements algebraic over the prime field).• The degree of imperfection of K.• The first-order theory, in a suitable ω-sorted language, of the inverse system of Galois groups al(L/K) where L runs over all finite Galois extensions of K.They also showed that the theory of PAC fields is undecidable, by showing that any graph can be encoded in the absolute Galois group of some PAC field. It turns out that the absolute Galois group controls much of the behaviour of the PAC fields. I will give below some examples illustrating this phenomenon.

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.

On PAC and bounded substructures of a stable structure

2006 ◽  
Vol 71 (2) ◽  
pp. 460-472 ◽  
Author(s):  
Anand Pillay ◽  
Dominika Polkowska
Keyword(s):  
Stable Structure ◽  
First Order ◽  
Closed Fields ◽  
Algebraically Closed Fields

AbstractWe introduce and study the notions of a PAC-substructure of a stable structure, and aboundedsubstructure of an arbitrary substructure, generalizing [10]. We give precise definitions and equivalences, saying what it means for properties such as PAC to be first order, study some examples (such as differentially closed fields) in detail, relate the material to generic automorphisms, and generalize a “descent theorem” for pseudo-algebraically closed fields to the stable context. We also point out that the elementary invariants of pseudo-algebraically closed fields from [6] are also valid for pseudo-differentially closed fields.

scholarly journals Coding complete theories in Galois groups

2008 ◽  
Vol 73 (2) ◽  
pp. 474-491
Author(s):  
James Gray
Keyword(s):  
Galois Groups ◽  
Vietoris Topology ◽  
Closed Fields ◽  
Prime Fields ◽  
Complete Theories ◽  
Pseudofinite Fields ◽  
Algebraically Closed Fields

AbstractIn this paper, I will give a new characterisation of the spaces of complete theories of pseudofinite fields and of algebraically closed fields with a generic automorphism (ACFA) in terms of the Vietoris topology on absolute Galois groups of prime fields.

An ω1-categorical ring which is not almost strongly minimal

1974 ◽  
Vol 39 (4) ◽  
pp. 665-668 ◽  
Author(s):  
K.-P. Podewski ◽  
J. Reineke
Keyword(s):  
Commutative Ring ◽  
Order Logic ◽  
First Order Logic ◽  
Complex Numbers ◽  
First Order ◽  
Factor Ring ◽  
Closed Fields ◽  
Ring Of Polynomials ◽  
Algebraically Closed Fields ◽  
The Ideal

A very important example of almost strongly minimal theories are the algebraically closed fields. A. Macintyre has shown [3] that every ω1-categorical field is algebraically closed. Therefore every ω1-categorical field is almost strongly minimal. It will be shown that not every ω1-categorical ring is almost strongly minimal.Let R0 be the factor ring C[y/(y2), where C[y] is the ring of polynomials in the indeterminate y over the field of complex numbers and (y2) the ideal generated by y2 in C[y].It is straightforward to prove that R0 has the following properties:1. R0 is a commutative ring with identity.2. R0 is of characteristic 0.3. For every polynomial p(x) = ∑ a1x1 ∈ R0[x] with of ai2 ≠ 0 for some i > 0 there is an a ∈ R0 such that p(a) · p(a) = 0.4. For all x, y ∈ R0 such that x2 = 0 and y ≠ 0 there exists a z ∈ R0 with y · z = x.5. There is an x ≠ 0 such that x2 = 0.These properties can be ∀∃-axiomatised in a countable first order logic (see [4]). Let T be the set of these sentences. With Theorem 7 we get that T is model-complete.If R is a model of T then I shall denote {a ∈ R ∣ a2 = 0}.

RAISING TO POWERS IN ALGEBRAICALLY CLOSED FIELDS

2003 ◽  
Vol 03 (02) ◽  
pp. 217-238 ◽  
Author(s):  
B. ZILBER
Keyword(s):  
Characteristic Zero ◽  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Closed Fields ◽  
Algebraically Closed Fields

We study structures on the fields of characteristic zero obtained by introducing (multi-valued) operations of raising to power. Using Hrushovski–Fraisse construction we single out among the structures exponentially-algebraically closed once and prove, under certain Diophantine conjecture, that the first order theory of such structures is model complete and every its completion is superstable.

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