Rich models

1990 ◽  
Vol 55 (3) ◽  
pp. 1292-1298 ◽  
Author(s):  
Michael H. Albert ◽  
Rami P. Grossberg
Keyword(s):  
Elementary Substructure ◽  
Uncountable Cardinal ◽  
Superstable Theories ◽  
Countable Theory

AbstractWe define a rich model to be one which contains a proper elementary substructure isomorphic to itself. Existence, nonstructure, and categoricity theorems for rich models are proved. A theory T which has fewer than min(2λ, ℶ2) rich models of cardinality λ (λ > ∣T∣) is totally transcendental. We show that a countable theory with a unique rich model in some uncountable cardinal is categorical in ℵ1 and also has a unique countable rich model. We also consider a stronger notion of richness, and use it to characterize superstable theories.

Some remarks on nonmultidimensional superstable theories

1994 ◽  
Vol 59 (1) ◽  
pp. 151-165 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
General Theory ◽  
Finite Rank ◽  
Atomic Model ◽  
Elementary Substructure ◽  
Finite Dimensional ◽  
Definable Groups ◽  
Superstable Theories ◽  
Finite Dimensional Case ◽  
Language Context ◽  
Minimal Formula

In this paper we study nonmultidimensional superstable theories T, possibly in an uncountable language, and develop some techniques permitting the generalisation of certain results from the finite rank (and/or countable language) context to the general case.We prove, among other things, the following: there is a set A0 of parameters, which has cardinality at most ∣T∣, and in the finite-dimensional case is finite, such that over any B ⊇ A0 there is a locally atomic model. One of the consequences of this is that if C is the monster model of T, φ(x) is a formula over A0, φC ⊇ X and (X, φC) satisfies the Tarski-Vaught condition after adding names for A0, then there is an elementary substructure M of C containing A0 such that φM = X. Applications to the spectrum problem will appear in [Ch-P].In fact, all the components of the machinery we develop are already present in the general theory. One such component involves a stratification of the regular types of T using a generalized notion of weakly minimal formula. This appears in [Sh, Chapter V and the proof of IX.2.4] and also in [P2]. A second component involves definable groups which arise as ‘binding” groups. The existence of such groups, under certain hypotheses on the behavior of nonorthogonality, is due to Hrushovski [Hr1], and our use of them to help obtain “j-constructible” models is similar to their use in [Bu-Sh].

Co-stationarity of the ground model

2006 ◽  
Vol 71 (3) ◽  
pp. 1029-1043 ◽  
Author(s):  
Natasha Dobrinen ◽  
Sy-David Friedman
Keyword(s):  
Large Cardinals ◽  
Partial Ordering ◽  
Proper Class ◽  
Ground Model ◽  
Covering Theorem ◽  
Uncountable Cardinal ◽  
Cohen Forcing ◽  
Measurable Cardinals

AbstractThis paper investigates when it is possible for a partial ordering ℙ to force Pk(Λ)\V to be stationary in Vℙ. It follows from a result of Gitik that whenever ℙ adds a new real, then Pk(Λ)\V is stationary in Vℙ for each regular uncountable cardinal κ in Vℙ and all cardinals λ ≥ κ in Vℙ [4], However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The following is equiconsistent with a proper class of ω1-Erdős cardinals: If ℙ is ℵ1-Cohen forcing, then Pk(Λ)\V is stationary in Vℙ, for all regular κ ≥ ℵ2and all λ ≩ κ. The following is equiconsistent with an ω1-Erdős cardinal: If ℙ is ℵ1-Cohen forcing, then is stationary in Vℙ. The following is equiconsistent with κ measurable cardinals: If ℙ is κ-Cohen forcing, then is stationary in Vℙ.

The spectrum of resplendency

1990 ◽  
Vol 55 (2) ◽  
pp. 626-636
Author(s):  
John T. Baldwin
Keyword(s):  
Homogeneous Model ◽  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Uncountable Cardinal ◽  
Recursive Theory ◽  
Finite Language

AbstractLet T be a complete countable first order theory and λ an uncountable cardinal. Theorem 1. If T is not superstable, T has 2λ resplendent models of power λ. Theorem 2. If T is strictly superstable, then T has at least min(2λ, ℶ2) resplendent models of power λ. Theorem 3. If T is not superstable or is small and strictly superstable, then every resplendent homogeneous model of T is saturated. Theorem 4 (with Knight). For each μ ∈ ω ∪ {ω, 2ω} there is a recursive theory in a finite language which has μ resplendent models of power κ for every infinite κ.

scholarly journals ON PRODUCTS OF ELEMENTARILY INDIVISIBLE STRUCTURES

2016 ◽  
Vol 81 (3) ◽  
pp. 951-971
Author(s):  
NADAV MEIR
Keyword(s):  
New Products ◽  
Elementary Substructure ◽  
First Order ◽  
Order Language ◽  
Image Position

AbstractWe say a structure ${\cal M}$ in a first-order language ${\cal L}$ is indivisible if for every coloring of its universe in two colors, there is a monochromatic substructure ${\cal M}\prime \subseteq {\cal M}$ such that ${\cal M}\prime \cong {\cal M}$. Additionally, we say that ${\cal M}$ is symmetrically indivisible if ${\cal M}\prime$ can be chosen to be symmetrically embedded in ${\cal M}$ (that is, every automorphism of ${\cal M}\prime$ can be extended to an automorphism of ${\cal M}$). Similarly, we say that ${\cal M}$ is elementarily indivisible if ${\cal M}\prime$ can be chosen to be an elementary substructure. We define new products of structures in a relational language. We use these products to give recipes for construction of elementarily indivisible structures which are not transitive and elementarily indivisible structures which are not symmetrically indivisible, answering two questions presented by A. Hasson, M. Kojman, and A. Onshuus.

Homogeneous changes in cofinalities with applications to HOD

2017 ◽  
Vol 17 (02) ◽  
pp. 1750007 ◽  
Author(s):  
Omer Ben-Neria ◽  
Spencer Unger
Keyword(s):  
Large Cardinals ◽  
Related Question ◽  
New Technique ◽  
Singular Cardinals ◽  
Uncountable Cardinal ◽  
A New Technique

We present a new technique for changing the cofinality of large cardinals using homogeneous forcing. As an application we show that many singular cardinals in [Formula: see text] can be measurable in HOD. We also answer a related question of Cummings, Friedman and Golshani by producing a model in which every regular uncountable cardinal [Formula: see text] in [Formula: see text] is [Formula: see text]-supercompact in HOD.

On the role of Ramsey quantifiers in first order arithmetic

1982 ◽  
Vol 47 (2) ◽  
pp. 423-435 ◽  
Author(s):  
James H. Schmerl ◽  
Stephen G. Simpson
Keyword(s):  
Formal System ◽  
Intended Meaning ◽  
Completeness Proof ◽  
First Order ◽  
Uncountable Cardinal ◽  
Consistent Extension ◽  
Witness Set ◽  
Order Arithmetic ◽  
Infinite Set

The purpose of this paper is to study a formal system PA(Q2) of first order Peano arithmetic, PA, augmented by a Ramsey quantifier Q2 which binds two free variables. The intended meaning of Q2xx′φ(x, x′) is that there exists an infinite set X of natural numbers such that φ(a, a′) holds for all a, a′ Є X such that a ≠ a′. Such an X is called a witness set for Q2xx′φ(x, x′). Our results would not be affected by the addition of further Ramsey quantifiers Q3, Q4, …, Here of course the intended meaning of Qkx1 … xkφ(x1,…xk) is that there exists an infinite set X such that φ(a1…, ak) holds for all k-element subsets {a1, … ak} of X.Ramsey quantifiers were first introduced in a general model theoretic setting by Magidor and Malitz [13]. The system PA{Q2), or rather, a system essentially equivalent to it, was first defined and studied by Macintyre [12]. Some of Macintyre's results were obtained independently by Morgenstern [15]. The present paper is essentially self-contained, but all of our results have been directly inspired by those of Macintyre [12].After some preliminaries in §1, we begin in §2 by giving a new completeness proof for PA(Q2). A by-product of our proof is that for every regular uncountable cardinal k, every consistent extension of PA(Q2) has a k-like model in which all classes are definable. (By a class we mean a subset of the universe of the model, every initial segment of which is finite in the sense of the model.)

scholarly journals Distributive and related ideals in generic extensions

1987 ◽  
Vol 106 ◽  
pp. 91-100
Author(s):  
C. A. Johnson
Keyword(s):  
Quotient Algebra ◽  
Complete Ideal ◽  
Naturally Occurring ◽  
Uncountable Cardinal ◽  
Game Theoretic ◽  
Generic Extensions

Let κ: be a regular uncountable cardinal and I a κ-complete ideal on te. In [11] Kanai proved that the μ-distributivity of the quotient algebra P(κ)I is preserved under κ-C.C. μ-closed forcing. In this paper we extend Kanai’s result and also prove similar preservation results for other naturally occurring forms of distributivity. We also consider the preservation of two game theoretic properties of I and in particular, using a game theoretic equivalent of precipitousness we give a new proof of Kakuda’s theorem ([10]) that the precipitousness of I is preserved under κ-C.C. forcing.

scholarly journals NONCONGLOMERABILITY FOR COUNTABLY ADDITIVE MEASURES THAT ARE NOT κ-ADDITIVE

2016 ◽  
Vol 10 (2) ◽  
pp. 284-300 ◽  
Author(s):  
MARK J. SCHERVISH ◽  
TEDDY SEIDENFELD ◽  
JOSEPH B. KADANE
Keyword(s):  
Conditional Probability ◽  
Inaccessible Cardinal ◽  
Conditional Probabilities ◽  
Uncountable Cardinal ◽  
Additive Probability ◽  
De Finetti ◽  
Additive Measures

AbstractLet κ be an uncountable cardinal. Using the theory of conditional probability associated with de Finetti (1974) and Dubins (1975), subject to several structural assumptions for creating sufficiently many measurable sets, and assuming that κ is not a weakly inaccessible cardinal, we show that each probability that is not κ-additive has conditional probabilities that fail to be conglomerable in a partition of cardinality no greater than κ. This generalizes a result of Schervish, Seidenfeld, & Kadane (1984), which established that each finite but not countably additive probability has conditional probabilities that fail to be conglomerable in some countable partition.

DETERMINACY AND JÓNSSON CARDINALS INL(ℝ)

2014 ◽  
Vol 79 (4) ◽  
pp. 1184-1198 ◽  
Author(s):  
S. JACKSON ◽  
R. KETCHERSID ◽  
F. SCHLUTZENBERG ◽  
W. H. WOODIN
Keyword(s):  
Uncountable Cardinal ◽  
Jonsson Cardinals

AbstractAssume ZF + AD +V=L(ℝ) and letκ< Θ be an uncountable cardinal. We show thatκis Jónsson, and that if cof (κ) = ω thenκis Rowbottom. We also establish some other partition properties.

Notes on the stability of separably closed fields

1979 ◽  
Vol 44 (3) ◽  
pp. 412-416 ◽  
Author(s):  
Carol Wood
Keyword(s):  
Differential Algebra ◽  
Complete Theory ◽  
Model Completeness ◽  
Closed Fields ◽  
Superstable Theories ◽  
Basic Facts ◽  
The Stability ◽  
Differential Fields

AbstractThe stability of each of the theories of separably closed fields is proved, in the manner of Shelah's proof of the corresponding result for differentially closed fields. These are at present the only known stable but not superstable theories of fields. We indicate in §3 how each of the theories of separably closed fields can be associated with a model complete theory in the language of differential algebra. We assume familiarity with some basic facts about model completeness [4], stability [7], separably closed fields [2] or [3], and (for §3 only) differential fields [8].

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