- Pittsburgh, Pennsylvania, United States
Rami Grossberg
Carnegie Mellon University, Mathematics, Faculty Member
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We study several cardinal, and ordinal--valued functions that are relatives of Hanf numbers. Let kappa be an infinite cardinal, and let T subseteq L_{kappa^+, omega} be a theory of cardinality <= kappa, and let gamma be an ordinal >=... more
We study several cardinal, and ordinal--valued functions that are relatives of Hanf numbers. Let kappa be an infinite cardinal, and let T subseteq L_{kappa^+, omega} be a theory of cardinality <= kappa, and let gamma be an ordinal >= kappa^+. For example we look at (1) mu_{T}^*(gamma, kappa):= min {mu^* for all phi in L_{infinity, omega}, with rk(phi)< gamma, if T has the (phi, mu^*)-order property then there exists a formula phi'(x;y) in L_{kappa^+, omega}, such that for every chi >= kappa, T has the (phi', chi)-order property}; and (2) mu^*(gamma, kappa):= sup{mu_T^*(gamma, kappa)| T in L_{kappa^+,omega}}.
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Complete and self-contained proofs to theorems of Shelah as well as some new results are presented: Theorem 0.1. Let Abe a set of formulas closed under Boolean operations and let p be a finite set of formulas. Ifp is A-stable then for A... more
Complete and self-contained proofs to theorems of Shelah as well as some new results are presented: Theorem 0.1. Let Abe a set of formulas closed under Boolean operations and let p be a finite set of formulas. Ifp is A-stable then for A > ( 2 ' T ' ) + + we have Theorem 0.2. R\p, A, A] = R\p, A, oo] for A > (2'')+. When R\p, A, oo] < oo then R\p, A, oo] < \T\. Theorem 0.1 is a generalization of Th II 3.11 from [Sha]. The proofs use the concepts of p is A-stable and p is A-superstable; which appear only implicitly in Shelah's work. We use the above functions to characterize p is A-superstable (when p — {x = x} and A = L, p is A-superstable iff T is superstable). The proofs presented here are simpler than Shelah's original presentation. We still use Shelah's ideas but in a different form. Let 51 be the "geometric" rank from the ACFA paper of Chatzidakis and Hrushovski. Theorem 0.3. Let p be a finite type. Ifp is stable then Sl\p] = D\p, L, oo] = R\p, L, oo].
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We want to show existence of indiscernible sets in models, without assuming the theory of the model is stable. Among other things we prove the following theorem: Let M be a model, and let X be a cardinal satisfying }J^=X. If fi does not... more
We want to show existence of indiscernible sets in models, without assuming the theory of the model is stable. Among other things we prove the following theorem: Let M be a model, and let X be a cardinal satisfying }J^=X. If fi does not have the a>-order property then for every ACM, |A|<X, and every ICM of cardinality X there exists JCI of cardinality X which is an indiscernible set over A. This is an improvement of a result of S. Shelah. Partially supported by the NSF. I would like to thank Michael Albert, and John Baldwin for suggestions to improvement of presentation.
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The aim of this paper is to set the foundation to separate geometric model theory from model theory. Our thesis is that it is possible to lift results from geometric model theory to non first order logic (e.g. LWl)W). We introduce a... more
The aim of this paper is to set the foundation to separate geometric model theory from model theory. Our thesis is that it is possible to lift results from geometric model theory to non first order logic (e.g. LWl)W). We introduce a relation between subsets of a pregeometry and show that it satisfies all the formal properties that forking satisfies in simple first order theories. This is important when one is trying to lift forking to nonelementary classes, in contexts where there exists pregeometries but not necessarily a well-behaved dependence relation (see for example [HySh]). We use these to reproduce S. Buechler's characterization of local modularity in general. These results are used by Lessmann to prove an abstract group configuration theorem in [Le2].
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Mathematics Technical Report
In [9] and [12], Shelah defined a certain type of Scott sentence which he called excellent. He proved, among other things, that if a Scott sentence is excellent and categorical in some uncountable power then it is categorical in all... more
In [9] and [12], Shelah defined a certain type of Scott sentence which he called excellent. He proved, among other things, that if a Scott sentence is excellent and categorical in some uncountable power then it is categorical in all uncountable powers: the analog of the Morley categoricity theorem. Proving such an analog is often the starting point in the classification of a family of classes. Before beginning this classification in the case of excellent Scott sentences, let us say a few words about what this paper is and what it is not.It is not the beginning of a classification theory for complete sentences in where is countable. Although excellence arises in the study of the model theory of Scott sentences, it is not a dividing line in a classification of them. In particular, the assumption of nonexcellence does not yield much information. In fact, in [3] there is an example of a nonexcellent Scott sentence, categorical in ℵ1 which is. not fully categorical. It seems to the second author that a classification of sentences analogous to the classification of first order theories is a long way off and may not be accomplishable in ZFC.This is not to say that the study of excellent Scott sentences (or the class of models of such which we will call excellent classes) is unproductive. Besides its extreme usefulness in [12], Mekler and Shelah have shown that excellence plays a decisive role in the study of almost free algebras (see [7]). Moreover, as the class of ω-saturated models of an ω-stable theory is an example of an excellent class, the study of excellent classes is at least as difficult as the study of first order ω-stable theories.
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Publication View. 43428938. Prof. Rami Grossberg (Advisor), Prof. John Baldwin, Prof. James Cummings, Prof. Dana Scott. (2007). June Department Of,; Olivier Lessmann,; Prof Rami Grossberg (advisor,; Prof John Baldwin,; Prof James Cum,;... more
Publication View. 43428938. Prof. Rami Grossberg (Advisor), Prof. John Baldwin, Prof. James Cummings, Prof. Dana Scott. (2007). June Department Of,; Olivier Lessmann,; Prof Rami Grossberg (advisor,; Prof John Baldwin,; Prof James Cum,; Prof Dana Scott. Abstract. ...
Saturation is (μ, κ)-transferable in T if and only if there is an expansion T1 of T with |T1| = |T| such that if M is a μ-saturated model of T1 and |M| ≥ κ then the reduct M|L(T) is κ-saturated. We characterize theories which are... more
Saturation is (μ, κ)-transferable in T if and only if there is an expansion T1 of T with |T1| = |T| such that if M is a μ-saturated model of T1 and |M| ≥ κ then the reduct M|L(T) is κ-saturated. We characterize theories which are superstable without f.c.p., or without f.c.p. as, respectively those where saturation is (ℵ0, λ)-transferable or (κ(T), λ)-transferable for all λ. Further if for some μ ≥ |T|,2μ > μ+, stability is equivalent to for all μ ≥ |T|, saturation is (μ, 2μ)-transferable.
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Let M be a given model with similarity type L = L(M), and let L′ be any fragment of L∣L(M∣+,ω of cardinality ∣L(M)∣. We call N ≺ ML′-relatively saturated iff for every B ⊆ N of cardinality less than ∥N∥ every L′-type over B which is... more
Let M be a given model with similarity type L = L(M), and let L′ be any fragment of L∣L(M∣+,ω of cardinality ∣L(M)∣. We call N ≺ ML′-relatively saturated iff for every B ⊆ N of cardinality less than ∥N∥ every L′-type over B which is realized in M is realized in N. We discuss the existence of such submodels.The following are corollaries of the existence theorems.(1) If M is of cardinality at least ℶω1, and fails to have the ω order property, then there exists N ≺ M which is relatively saturated in M of cardinality ℶω1.(2) Assume GCH. Let ψ ∈ Lω1, ω, and let L′ ⊆ Lω1, ω be a countable fragment containing ψ. If ∃χ > ℵ0 such that I(χ, ψ) < 2χ, then for every M ⊨ ψ and every cardinal λ < ∥M∥ of uncountable cofinality, M has an L′-relatively saturated submodel of cardinality λ.
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Let L(Q) be first order logic with Keisler's quantifier, in the λ+ interpretation (= the satisfaction is defined as follows: M ⊨ (Qx)φ(x) means there are λ+ many elements in M satisfying the formula φ(x)).Theorem 1. Let λ be a... more
Let L(Q) be first order logic with Keisler's quantifier, in the λ+ interpretation (= the satisfaction is defined as follows: M ⊨ (Qx)φ(x) means there are λ+ many elements in M satisfying the formula φ(x)).Theorem 1. Let λ be a singular cardinal; assume □λ and GCH. If T is a complete theory in L(Q) of cardinality at most λ, and p is an L(Q) 1-type so that T strongly omits p( = p has no support, to be defined in §1), then T has a model of cardinality λ+ in the λ+ interpretation which omits p.Theorem 2. Let λ be a singular cardinal, and let T be a complete first order theory of cardinality λ at most. Assume □λ and GCH. If Γ is a smallness notion then T has a model of cardinality λ+ such that a formula φ(x) is realized by λ+ elements of M iff φ(x) is not Γ-small. The theorem is proved also when λ is regular assuming λ = λ
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Let κ and λ be infinite cardinals such that λ ≤ λ (we have new information for the case when κ ≤ λ). Let T be a theory in Lκ +, ω of cardinality at most κ, let . Now defineOur main concept in this paper is is a theory in Lκ +, ω of... more
Let κ and λ be infinite cardinals such that λ ≤ λ (we have new information for the case when κ ≤ λ). Let T be a theory in Lκ +, ω of cardinality at most κ, let . Now defineOur main concept in this paper is is a theory in Lκ +, ω of cardinality κ at most, and φ(x, y) ϵ Lκ +, ω}. This concept is interesting because ofTheorem 1. Let T ⊆ Lκ +, ω of cardinality ≤ κ, and. Ifthen (∀χ > κ)I(χ, T) = 2χ (where I(χ, T) stands for the number of isomorphism types of models of T of cardinality χ).Many years ago the second author proved that . Here we continue that work by provingTheorem 2. .Theorem 3. For everyκ ≤ λwe have.For some κ or λ we have better bounds than in Theorem 3, and this is proved via a new two cardinal theorem.Theorem 4. For every T ⊆ Lκ +, ω, and any set of formulas ⊆ Lκ +, ω such thatT ⊇ Lκ +, ω, if T is (, μ)-unstable for μ satisfyingμμ*(λ,κ) = μ then T is-unstable (i.e. for every χ ≥ λ, T is (, χ)-unstable). Moreover, T is Lκ +, ω-unstable.In the second part of the paper,...
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Abstract: "We continue the study of stability of a type in several directions: (1) Inside a fixed model, (2) for classes of models where the compactness theorem fails and (3) for the first order case. Appropriate localizations of the... more
Abstract: "We continue the study of stability of a type in several directions: (1) Inside a fixed model, (2) for classes of models where the compactness theorem fails and (3) for the first order case. Appropriate localizations of the order property, the independence property, and the strict order property are introduced. We are able to generalize some of the results that were known in the case of local stability for the first order theories, and for stability for nonelementary classes (existence of indiscernibles, existence of averages, stability spectrum, equivalence between order and instability). In the first order case, we also prove the local version of Shelah's Trichotomy Theorem. Finally as an application, we give a new characterization of stable types when the ambient first order theory is simple."
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We prove that several definitions of superstability in abstract elementary classes (AECs) are equivalent under the assumption that the class is stable, tame, has amalgamation, joint embedding, and arbitrarily large models. This partially... more
We prove that several definitions of superstability in abstract elementary classes (AECs) are equivalent under the assumption that the class is stable, tame, has amalgamation, joint embedding, and arbitrarily large models. This partially answers questions of Shelah. <strong>Theorem 0.1</strong>. Let K be a tame AEC with amalgamation, joint embedding, and arbitrarily large models. Assume K is stable. Then the following are equivalent: (1) For all high-enough λ, there exists κ ≤ λ such that there is a good λ-frame on the class of κ-saturated models in Kλ. (2) For all high-enough λ, K has a unique limit model of cardinality λ. (3) For all high-enough λ, K has a superlimit model of cardinality λ. (4) For all high-enough λ, the union of a chain of λ-saturated models is λ-saturated. (5) There exists θ such that for all high-enough λ, K is (λ, θ)- solvable.
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Abstract: "The aim of this paper is to make progress towards a geometric model theory for non first order theories. The main difficulty is to work in an environment where the compactness theorem fails. This paper continues the work... more
Abstract: "The aim of this paper is to make progress towards a geometric model theory for non first order theories. The main difficulty is to work in an environment where the compactness theorem fails. This paper continues the work started in [GrLe1]. The main result is an axiomatic approach to the Hrushovski-Zilber group configuration theorem."
In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and... more
In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent: Corollary Let K be a tame AEC with a monster model. Assume that K is stable in a proper class of cardinals. The following are equivalent: 1) For all high-enough λ, K has no long splitting chains. 2) For all high-enough λ, there exists a good λ-frame on a skeleton of K_λ. 3) For all high-enough λ, K has a unique limit model of cardinality λ. 4) For all high-enough λ, K has a superlimit model of cardinality λ. 5) For all high-enough λ, th...
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Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstability-like property for a certain independence relation called... more
Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstability-like property for a certain independence relation called nonsplitting. We generalize their result as follows: given an abstract notion of independence for Galois (orbital) types over models, we derive that the notion satisfies a superstability property provided that the class is categorical and satisfies a weakening of amalgamation. This extends the Shelah-Villaveces result (the independence notion there was splitting) as well as a result of the first and second author where the independence notion was coheir. The argument is in ZFC and fills a gap in the Shelah-Villaveces proof.