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Classification Theory for Abstract Elementary Classes
 In Logic and Algebra, Yi Zhang editor, Contemporary Mathematics 302, AMS,(2002), 165–203
, 2002
"... In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the firstorder case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. S ..."
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In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the firstorder case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. Some of the most basic results in classification theory for non elementary classes are presented. The motivating point of view is Shelah's categoricity conjecture for L# 1 ,# . While only very basic theorems are proved, an effort is made to present number of different technologies: Flavors of weak diamond, models of weak set theories, and commutative diagrams. We focus in issues involving existence of Galois types, extensions of types and Galoisstability.
A five element basis for the uncountable linear orders
 Annals of Mathematics
"... Abstract. In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five element basis. In fact such a basis f ..."
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Cited by 11 (3 self)
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Abstract. In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five element basis. In fact such a basis follows from the Proper Forcing Axiom, a strong form of the Baire Category Theorem. The elements are X, ω1, ω ∗ 1, C, C ∗ where X is any suborder of the reals of cardinality ℵ1 and C is any Countryman line. This confirms a longstanding conjecture of Shelah. 1.
Shelah’s categoricity conjecture from a successor for tame abstract elementary classes
 The Journal of Symbolic Logic
, 2006
"... elementary classes. Theorem 0.1. Suppose that K is a χtame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ, LS(K) +}. If K is categorical in λ and λ +, then K is categorical in λ ++. Combining this theorem with ..."
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Cited by 9 (3 self)
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elementary classes. Theorem 0.1. Suppose that K is a χtame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ, LS(K) +}. If K is categorical in λ and λ +, then K is categorical in λ ++. Combining this theorem with some results from [Sh 394], we derive a form of Shelah’s Categoricity Conjecture for tame abstract elementary classes: Corollary 0.2. Suppose K is a χtame abstract elementary class satisfying the amalgamation and joint embedding properties. Let µ0:= Hanf(K). If χ ≤ ℶ (2 µ 0) + and K is categorical in some λ +> ℶ (2 µ 0) +, then K is categorical in µ for all µ> ℶ (2 µ 0) +.
A primer of simple theories
 Archive Math. Logic
"... Abstract. We present a selfcontained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelah’s 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. ..."
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Cited by 9 (2 self)
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Abstract. We present a selfcontained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelah’s 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. The exposition is from a contemporary perspective and takes into account contributions due to S. Buechler, E. Hrushovski, B. Kim, O. Lessmann, S. Shelah and A. Pillay.
Independence in finitary abstract elementary classes
 Ann. Pure Appl. Logic
, 2006
"... In this paper we study a specific subclass of abstract elementary classes. We construct a notion of independence for these AEC’s and show that under simplicity the notion has all the usual properties of first order nonforking over complete types. Our approach generalizes the context of ℵ0stable ho ..."
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Cited by 5 (0 self)
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In this paper we study a specific subclass of abstract elementary classes. We construct a notion of independence for these AEC’s and show that under simplicity the notion has all the usual properties of first order nonforking over complete types. Our approach generalizes the context of ℵ0stable homogeneous classes and excellent classes. Our set of assumptions follow from disjoint amalgamation, existence of a prime model over ∅, LöwenheimSkolem number being ω, LS(K)tameness and a property we call finite character. We also start the studies of these classes from the ℵ0stable case. Stability in ℵ0 and LS(K)tameness can be replaced by categoricity above the Hanf number. Finite character is the main novelty of this paper. Almost all examples of AEC’s have this property and it allows us to use weak types, as we call them, in place of Galois types. 1
Properness without elementaricity
, 2003
"... We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary submodels of some (H(χ), ∈). This leads to forcing notions which are “reasonably ” definable. We present two specific properties materializing ..."
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Cited by 5 (2 self)
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We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary submodels of some (H(χ), ∈). This leads to forcing notions which are “reasonably ” definable. We present two specific properties materializing this intuition: nep (nonelementary properness) and snep (Souslin nonelementary properness) and also the older Souslin proper. For this we consider candidates (countable models to which the definition applies). A major theme here is “preservation by iteration”, but we also show a dichotomy: if such forcing notions preserve the positiveness of the set of old reals for some naturally defined c.c.c. ideal, then they preserve the positiveness of any old positive set hence preservation by composition of two follows. Last but not least, we prove that (among such forcing notions) the only one commuting with Cohen is Cohen itself; in other words, any other such forcing notion make the set of old reals to a meager set. In the end we present some open problems in this area.
Constructing Boolean algebras for cardinal invariants
, 2001
"... We construct Boolean Algebras answering some questions of J. Donald Monk on cardinal invariants. The results are proved in ZFC (rather than giving consistency results). We deal with the existence of superatomic Boolean Algebras with “few automorphisms”, with entangled sequences of linear orders, a ..."
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Cited by 4 (3 self)
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We construct Boolean Algebras answering some questions of J. Donald Monk on cardinal invariants. The results are proved in ZFC (rather than giving consistency results). We deal with the existence of superatomic Boolean Algebras with “few automorphisms”, with entangled sequences of linear orders, and with semiZFC examples of the nonattainment of the spread (and hL, hd).
An introduction to excellent classes
 Logic and its Applications, Contemporary Mathematics
, 2005
"... ABSTRACT. In this paper, we present Shelah’s theory of excellence concluding with his categoricity theorem for the class of atomic models of a countable first order theory, under the assumption that there exists a large full model. This ..."
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ABSTRACT. In this paper, we present Shelah’s theory of excellence concluding with his categoricity theorem for the class of atomic models of a countable first order theory, under the assumption that there exists a large full model. This
CCC forcings and splitting reals
 Israel Journal of Mathematics
"... Abstract. Prikry asked if it is relatively consistent with the usual axioms of ZFC that every nontrivial ccc forcing adds either a Cohen or a random real. Both Cohen and random reals have the property that they neither contain nor are disjoint from an infinite set of integers in the ground model, i. ..."
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Abstract. Prikry asked if it is relatively consistent with the usual axioms of ZFC that every nontrivial ccc forcing adds either a Cohen or a random real. Both Cohen and random reals have the property that they neither contain nor are disjoint from an infinite set of integers in the ground model, i.e. they are splitting reals. In this note I show that that it is relatively consistent with ZFC that every non atomic weakly distributive ccc forcing adds a splitting real. This holds, for instance, under the Proper Forcing Axiom and is proved using the Pideal dichotomy first formulated by Abraham and Todorcevic [AT] and later extended by Todorcevic [T]. In the process, I show that under the Pideal dichotomy every weakly distributive ccc complete Boolean algebra carries an exhaustive submeasure, a result which has some interest in its own right. Using a previous theorem of Shelah [Sh1] it follows that a modified Prikry conjecture holds in the context of Souslin forcing notions, i.e. every non atomic ccc Souslin forcing either adds a Cohen real or its regular open algebra is a Maharam algebra. 1.
CATEGORICITY, AMALGAMATION, AND TAMENESS
"... ABSTRACT. Theorem. For each 2 ≤ k < ω there is an Lω1,ωsentence φk such that: (1) φk is categorical in µ if µ ≤ ℵk−2; (2) φk is not ℵk−2Galois stable; (3) φk is not categorical in any µ with µ> ℵk−2; (4) φk has the disjoint amalgamation property; (5) For k> 2, (a) φk is (ℵ0, ℵk−3)tame; indeed, sy ..."
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ABSTRACT. Theorem. For each 2 ≤ k < ω there is an Lω1,ωsentence φk such that: (1) φk is categorical in µ if µ ≤ ℵk−2; (2) φk is not ℵk−2Galois stable; (3) φk is not categorical in any µ with µ> ℵk−2; (4) φk has the disjoint amalgamation property; (5) For k> 2, (a) φk is (ℵ0, ℵk−3)tame; indeed, syntactic firstorder types determine Galois types over models of cardinality at most ℵk−3; (b) φk is ℵmGalois stable for m ≤ k − 3; (c) φk is not (ℵk−3, ℵk−2)tame. We adapt an example of [9]. The amalgamation, tameness, stability results, and the contrast between syntactic and Galois types are new; the categoricity results refine the earlier work of Hart and Shelah and answer a question posed by Shelah in [17].