Set Of Indiscernibles Research Articles

Saturated free algebras and almost indiscernible theories

We extend the concept of “almost indiscernible theory” introduced by Pillay and Sklinos in 2015 (which was itself a modernization and expansion of Baldwin and Shelah from 1983), to uncountable languages and uncountable parameter sequences. Roughly speaking a theory $$T$$ is almost indiscernible if some saturated model is in the algebraic closure of an indiscernible set of sequences. We show that such a theory $$T$$ is nonmultidimensional, superstable, and stable in all cardinals $$\ge |T|$$ . We prove a structure theorem for sufficiently large $$ a $$ -models $$M$$ , which states that over a suitable base, $$M$$ is in the algebraic closure of an independent set of realizations of weight one types (in possibly infinitely many variables). We also explore further the saturated free algebras of Baldwin and Shelah in both the countable and uncountable context. We study in particular theories and varieties of $$R$$ -modules, characterizing those rings $$ R $$ for which the free $$ R $$ -module on $$ \left| R\right| ^{+} $$ generators is saturated, and pointing out a counterexample to a conjecture by Pillay and Sklinos.

An unpublished theorem of Solovay on OD partitions of reals into two non-OD parts, revisited

A definable pair of disjoint non-OD sets of reals (hence, indiscernible sets) exists in the Sacks and 𝔼0-large generic extensions of the constructible universe L. More specifically, if [Formula: see text] is either Sacks generic or 𝔼0-large generic real over L, then it is true in L[a] that there is a lightface [Formula: see text] equivalence relation [Formula: see text] on the [Formula: see text] set [Formula: see text] with exactly two equivalence classes, and both those classes are non-OD sets.

A set-theoretic approach to observability and its application to process control

In this paper, a new approach to the observability of dynamical systems with uncertain sensors is developed. This approach is based on the redefinition of indistinguishability to consider measurement uncertainty and on the definition of a set called the final indiscernible set, which is directly related to state estimation accuracy and whose size is a quantitative observability index. Making use of this index, an observability-oriented controller, which attempts to drive the system through a high-observability trajectory to reduce estimation error, is proposed. Finally, this controller is illustrated through its application to the model of a CSTR, showing that the proposed method enhances the accuracy of state estimation when compared with a similar control scheme that does not consider observability and with a another kind of controller that employs a different observability index.

SATURATED FREE ALGEBRAS REVISITED

AbstractWe give an exposition of results of Baldwin–Shelah [2] on saturated free algebras, at the level of generality of complete first order theories T with a saturated model M which is in the algebraic closure of an indiscernible set. We then make some new observations when M is a saturated free algebra, analogous to (more difficult) results for the free group, such as a description of forking.

Indiscernibles, EM-Types, and Ramsey Classes of Trees

The author has previously shown that for a certain class of structures I , I -indexed indiscernible sets have the modeling property just in case the age of I is a Ramsey class. We expand this known class of structures from ordered structures in a finite relational language to ordered, locally finite structures which isolate quantifier-free types by way of quantifier-free formulas. This result is applied to give new proofs that certain classes of trees are Ramsey. To aid this project we develop the logic of EM-types.

Regularity lemmas for stable graphs

We develop a framework in which Szemeredi's celebrated Regularity Lemma for graphs interacts with core model-theoretic ideas and techniques. Our work relies on a coincidence of ideas from model theory and graph theory: arbitrarily large half-graphs coincide with model-theoretic instability, so in their absence, structure theorems and technology from stability theory apply. In one direction, we address a problem from the classical Szemeredi theory. It was known that the pairs in the statement of Szemeredi's regularity lemma cannot be eliminated, due to the counterexample of half-graphs (i.e., the order property, corresponding to model-theoretic instability). We show that half-graphs are the only essential difficulty, by giving a much stronger version of Szemeredi's regularity lemma for models of stable theories of graphs (i.e. graphs with the non-k�-order property), in which there are no irregular pairs, the bounds are significantly improved, and each component satisfies an condition. In the other direction, we take a more model-theoretic approach, and give several new Szemeredi-type partition theorems for models of stable theories of graphs. The first theorem gives a partition of any such graph into indiscernible components, meaning here that each component is either a complete or an empty graph, whose interaction is strongly uniform. This relies on a finitary version of the classic model-theoretic fact that stable theories admit large sets of indiscernibles, by showing that in models of stable theories of graphs one can extract much larger indiscernible sets than expected by Ramsey's theorem. The second and third theorems allow for a much smaller number of components at the cost of weakening the indivisibility condition on the components. We also discuss some extensions to graphs without the independence property. All graphs are finite and all partitions are equitable, i.e. the sizes of the components differ by at most 1. In the last three theorems, the number of components depends on the size of the graph; in the first theorem quoted, this number is a function ofonly as in the usual Szemeredi regularity lemma.

Characterization of NIP theories by ordered graph-indiscernibles

We generalize the Unstable Formula Theorem characterization of stable theories from Shelah (1978) [11], that a theory T is stable just in case any infinite indiscernible sequence in a model of T is an indiscernible set. We use a generalized form of indiscernibles from [11], in our notation, a sequence of parameters from an L-structure M, (bi:i∈I), indexed by an L′-structure I is L′-generalized indiscernible inM if qftpL′(i¯;I)=qftpL′(j¯;I) implies tpL(b¯i¯;M)=tpL(b¯j¯;M) for all same-length, finite i¯,j¯ from I. Let Tg be the theory of linearly ordered graphs (symmetric, with no loops) in the language with signature Lg={<,R}. Let Kg be the class of all finite models of Tg. We show that a theory T has NIP if and only if any Lg-generalized indiscernible in a model of T indexed by an Lg-structure with age equal to Kg is an indiscernible sequence.

Compact complex manifolds with the DOP and other properties

AbstractWe point out that a certain complex compact manifold constructed by Lieberman has the dimensional order property, and has U-rank different from Morley rank. We also give a sufficient condition for a Kähler manifold to be totally degenerate (that is, to be an indiscernible set, in its canonical language) and point out that there are K3 surfaces which satisfy these conditions.

Stability theory, permutations of indiscernibles, and embedded finite models

We show that the expressive power of first-order logic over finite models embedded in a model M M is determined by stability-theoretic properties of M M . In particular, we show that if M M is stable, then every class of finite structures that can be defined by embedding the structures in M M , can be defined in pure first-order logic. We also show that if M M does not have the independence property, then any class of finite structures that can be defined by embedding the structures in M M , can be defined in first-order logic over a dense linear order. This extends known results on the definability of classes of finite structures and ordered finite structures in the setting of embedded finite models. These results depend on several results in infinite model theory. Let I I be a set of indiscernibles in a model M M and suppose ( M , I ) (M,I) is elementarily equivalent to ( M 1 , I 1 ) (M_1,I_1) where M 1 M_1 is | I 1 | + |I_1|^+ -saturated. If M M is stable and ( M , I ) (M,I) is saturated, then every permutation of I I extends to an automorphism of M M and the theory of ( M , I ) (M,I) is stable. Let I I be a sequence of &gt; &gt; -indiscernibles in a model M M , which does not have the independence property, and suppose ( M , I ) (M,I) is elementarily equivalent to ( M 1 , I 1 ) (M_1,I_1) where ( I 1 , &gt; ) (I_1,&gt;) is a complete dense linear order and M 1 M_1 is | I 1 | + |I_1|^+ -saturated. Then ( M , I ) (M,I) -types over I I are order-definable and if ( M , I ) (M,I) is ℵ 1 \aleph _1 -saturated, every order preserving permutation of I I can be extended to a back-and-forth system.

Countably Categorical Structures with n-Degenerate Algebraic Closure

We study the class of ω-categorical structures with n-degenerate algebraic closure for some n ε ω, which includes ω-categorical structures with distributive lattice of algebraically closed subsets (see [4]), and in particular those with degenerate (trivial) algebraic closure. We focus on the models of ω-categorical universal theories, absolutely ubiquitous structures, and ω-categorical structures generated by an indiscernible set. The assumption of n-degeneracy implies total categoricity for the first class, stability for the second, and ω-stability for the third.

A remark on minimal models

We prove the following theorem: Let $T$ be superstable and let $A$ any set. Then there is no minimal model over $A$ which has an infinite set of indiscernibles over $A$.

On the existence of atomic models

AbstractWe give an example of a countable theory T such that for every cardinal λ ≥ ℵ2 there is a fully indiscernible set A of power λ such that the principal types are dense over A, yet there is no atomic model of T over A. In particular, T(A) is a theory of size λ where the principal types are dense, yet T(A) has no atomic model.

Indiscernible sequences in a model which fails to have the order property

AbstractBasic results on the model theory of substructures of a fixed model are presented. The main point is to avoid the use of the compactness theorem, so this work can easily be applied to the model theory of Lω1,ω and its relatives. Among other things we prove the following theorem:Let M be a model, and let λ be a cardinal satisfying λ∣L(M)∣ = λ. If M does not have the ω-order property, then for every A ⊆ M, ∣A∣ ≤ λ, and every I ⊆ M of cardinality λ+ there exists J ⊆ I cardinality λ+ which is an indiscernible set over A.This is an improvement of a result of S. Shelah.

Large resplendent models generated by indiscernibles

The motivation for the results presented here comes from the following two known theorems which concern countable, recursively saturated models of Peano arithmetic.(1) if is a countable, recursively saturated model of PA, then for each infinite cardinal κ there is a resplendent which has cardinality κ. (See Theorem 10 of [1].)(2) if is a countable, recursively saturated model of PA, then is generated by a set of indiscernibles. (See [4].)It will be shown here that (1) and (2) can be amalgamated into a common generalization.(3) if is a countable, recursively saturated model of PA, then for each infinite cardinal κ there is a resplendent which has cardinality κ and which is generated by a set of indiscernibles.By way of contrast we will also get recursively saturated models of PA which fail to be resplendent and yet are generated by indiscernibles.(4) if is a countable, recursively saturated model of PA, then for each uncountable cardinal κ there is a κ-like recursively saturated generated by a set of indiscernibles.None of (1), (2) or (3) is stated in its most general form. We will make some comments concerning their generalizations. From now on let us fix a finite language L; all structures considered are infinite L-structures unless otherwise indicated.

Large resplendent models generated by indiscernibles

The motivation for the results presented here comes from the following two known theorems which concern countable, recursively saturated models of Peano arithmetic.(1) if is a countable, recursively saturated model of PA, then for each infinite cardinal κ there is a resplendent which has cardinality κ. (See Theorem 10 of [1].)(2) if is a countable, recursively saturated model of PA, then is generated by a set of indiscernibles. (See [4].)It will be shown here that (1) and (2) can be amalgamated into a common generalization.(3) if is a countable, recursively saturated model of PA, then for each infinite cardinal κ there is a resplendent which has cardinality κ and which is generated by a set of indiscernibles.By way of contrast we will also get recursively saturated models of PA which fail to be resplendent and yet are generated by indiscernibles.(4) if is a countable, recursively saturated model of PA, then for each uncountable cardinal κ there is a κ-like recursively saturated generated by a set of indiscernibles.None of (1), (2) or (3) is stated in its most general form. We will make some comments concerning their generalizations. From now on let us fix a finite language L; all structures considered are infinite L-structures unless otherwise indicated.

Finitely based theories

AbstractA stable theory is finitely based if every set of indiscernibles is based on a finite subset. This is a common generalization of superstability and 1-basedness. We show that if such theories have more than one model they must have infinitely many, and prove some other conjectures.

Structures coordinatized by indiscernible sets

Etude des structures denombrables categoriques stables coordonnees par des ensembles indiscernables

Prikry forcing at κ+ and beyond

If U is a normal measure on κ then we can add indiscernibles for U either by Prikry forcing [P] or by taking an iterated ultrapower which will add a sequence of indiscernibles for over M. These constructions are equivalent: the set C of indiscernibles for added by the iterated ultrapower is Prikry generic for [Mat]. Prikry forcing has been extended for sequences of measures of length by Magidor [Mag], and his method readily extends to . In this case the measure U is replaced by a sequence of measures and the set C of indiscernibles is replaced by a system of indiscernibles for : is a function such that (κ, β) is a set of indiscernibles for (κ, β) for each . The equivalence between forcing and iterated ultra-powers still holds true for such sequences: there is an interated ultrapower j: V → M (which is defined in detail later in this paper) such that the system of indiscernibles for j() constructed by j is Magidor generic over M.The construction of the system of indiscernibles works equally well for o(κ) ≧ κ+. Radin has defined a variant of Prikry forcing which also works for o(κ) &gt; κ+ ([R]; see also [Mi82] where Radin forcing is applied specifically to sequences of measures, rather than to hypermeasures as in Radin's paper), but Radin's forcing is weaker than Magidor's extension of Prikry forcing in the sense that the system of indiscernibles generated by the interated ultrapower is not Radin generic for j(), but only the set . That is, an indiscernible does not belong to a specific measure, but only to the whole sequence of measures on the cardinal κ.

Degrees of Indiscernibles in Decidable Models

We show that the problem of finding an infinite set of indiscernibles in an arbitrary decidable model of a first order theory is essentially equivalent to the problem of finding an infinite path through a recursive $\omega$-branching tree. Similarly, we show that the problem of finding an infinite set of indiscernibles in a decidable model of an $\omega$-categorical theory with decidable atoms is essentially equivalent to finding an infinite path through a recursive binary tree.

Degrees of indiscernibles in decidable models

We show that the problem of finding an infinite set of indiscernibles in an arbitrary decidable model of a first order theory is essentially equivalent to the problem of finding an infinite path through a recursive ω \omega -branching tree. Similarly, we show that the problem of finding an infinite set of indiscernibles in a decidable model of an ω \omega -categorical theory with decidable atoms is essentially equivalent to finding an infinite path through a recursive binary tree.