Universal theories categorical in power and κ-generated models (original) (raw)
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On the existence of universal models
Archive for Mathematical Logic, 2004
Suppose that λ = λ <λ ≥ ℵ 0 , and we are considering a theory T. We give a criterion on T which is sufficient for the consistent existence of λ ++ universal models of T of size λ + for models of T of size ≤ λ + , and is meaningful when 2 λ + > λ ++. In fact, we work more generally with abstract elementary classes. The criterion for the consistent existence of universals applies to various well known theories, such as triangle-free graphs and simple theories. Having in mind possible applications in analysis, we further observe that for such λ, for any fixed µ > λ + regular with µ = µ λ + , it is consistent that 2 λ = µ and there is no normed vector space over Q of size < µ which is universal for normed vector spaces over Q of dimension λ + under the notion of embedding h which specifies (a, b) such that ||h(x)||/||x|| ∈ (a, b) for all x. 1 In the list of publications of S. Shelah, this is publication number 614. Both authors thank the United States-Israel Binational Science Foundation for a partial support and various readers of the manuscript. for their helpful comments.
Journal of Symbolic Logic, 1986
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,...
On a Classification of Theories Without the
2010
Abstract. A theory is stable up to ∆ if any ∆-type over a model has a few extensions up to complete types. I prove that a theory has no the independence property iff it is stable up to some ∆, where each ϕ(x; ¯y) ∈ ∆ has no the independence property. Definability of one-types over a model of a stable up to ∆ theory is investigated. 1.
ON THE SYMBIOSIS BETWEEN MODEL-THEORETIC AND SET-THEORETIC PROPERTIES OF LARGE CARDINALS
We study some large cardinals in terms of reflection, establishing new connections between the model-theoretic and the set-theoretic approaches. §1. Introduction. First-order logic alone cannot express important properties such as finiteness or uncountability of the model, well-foundedness of a binary predicate, completeness of a linear order, etc. This led to Mostowski [10] and later Lindström [6] to introduce the concept of a generalised quantifier. This made it possible to compare model-theoretic and set-theoretic definability of various mathematical concepts. It turned out that there is a close connection between the two. Following [14], we call this connection symbiosis.
Model-completeness in a first order language with a generalized quantifier
Pacific Journal of Mathematics, 1975
The concept of Model-Completeness is defined in a first order language with a generalized quantifier. A necessary and sufficient condition is given for that Model-Completeness and its relation to categoricity is discussed. Some results of this paper were obtained in the author's thesis [12] and were announced in [11], They, together with other results of [12] were improved independently by the author and by S. Shelah. A suggestion of S. Shelah made some proofs simpler and due to it, better results were obtained in Theorem 1.5. The author wishes to thank S. Shelah for his remarks. Let L be a first order language with equality and let L(Q) be the language obtained from L by adding a new quantifier Q. Let α, β denote infinite cardinals. We define α-satisfaction for L(Q) by interpreting Q as "there exist at least a elements". If a sentence φ of L(Q) is a-satisfied in a model 91 for L we write 9ll= α φ and we say that 91 is an a-model for φ. Let 31,93 be two models for L, |9l|^α and 91 C 93. Write 21 < a 93 if for every n, every formula φ{x u-,x n) in L(Q) and every a u-,a n in 9Ϊ: 9lM>[α,, ,αj iff 93h α φ[α,, ,α n ]. Let T be an ordinary first order theory (namely a theory in L) that has infinite models. Define T(Q) = TU{Qx[x =x]}. Call T a-model-complete if for every 21,93 which are α-models for T(Q) and 91 C 93 also 9l< α 93. A necessary and sufficient condition for T to be a-model-complete for a > H o is given in section 1. Let T be as before. Define T(a) = {φ: φ is a sentence in L(Q) and for every 91, if 9lh α T(Q) then 9ίh α φ}. Call T α-complete if for every sentence φ in L(Q) either φ G T(a) ory φ G Γ(α). In §2, it is shown that if T is categorical in one uncountable power, it is incomplete and for every a^H*: T(a)=T(Ho). If T is also modelcomplete (in the usual sense) then it is α-model-complete for every α^No and T(a) is decidable provided T is axiomatic. 1. ci-Model-Completeness. DEFINITION 1.1. Let φ(x,x u ,jc m) be a formula in L such that x, jt,, ,jc m are exactly all its free variables. Let 91 be a model for L and let α,, ,α m be elements in 91. Define: 265
Some Two-Cardinal Results for O-Minimal Theories
Journal of Symbolic Logic, 1998
We examine two-cardinal problems for the class of O-minimal theories. We prove that an O-minimal theory which admits some (κ, λ) must admit every (κ , λ ). We also prove that every "reasonable" variant of Chang's Conjecture is true for O-minimal structures. Finally, we generalize these results from the two-cardinal case to the δ-cardinal case for arbitrary ordinals δ.
Theories with a finite number of countable models
The Journal of Symbolic Logic, 1978
We give two examples. T0 has nine countable models and a nonprincipal 1-type which contains infinitely many 2-types. T1, has four models and an inessential extension T2 having infinitely many models.