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Set existence axioms for general (not necessarily countable) stability theory

Victor Harnik
1987 Annals of Pure and Applied Logic  
Set existence axioms for stability theory 233 bounded and we define 2;, II;, A; for i = 0, 1 and j E o as is customarily done in  ...  The aim of this paper is to extend our work [5] by formulating set existence axioms in a general context that can accommodate model theory of arbitrary (not necessarily countable) languages and then  ...  H owever, the distinction between the countable and the general (i.e., not necessarily countable) theory is not very fashionable nowadays, neither in algebra nor in parts of model theory because, except  ... 
doi:10.1016/0168-0072(87)90002-9 fatcat:bwe2es4qznahlasxgsg2rphfwi

Book Review: Stability in model theory

Victor Harnik
1989 Bulletin of the American Mathematical Society  
Nowadays, we prefer the shorter name of "w-stability." co-stable theories are not necessarily categorical in some uncountable power but have good algebraic properties.  ...  Notice that L and, hence, T are allowed to have not only countable but also uncountable cardinalities, a fact that increases the scope of model theory.  ... 
doi:10.1090/s0273-0979-1989-15781-9 fatcat:os4nxovvu5f37km2xkzvsplqru

Book Review: Fundamentals of stability theory

Gregory L. Cherlin
1989 Bulletin of the American Mathematical Society  
One may also look at more concrete contexts, arising for example in algebra, which are not necessarily special cases of the abstract context.  ...  Shelah has completed the full first order case for countable theories [12] , and this will be in the second edition of [11] .  ... 
doi:10.1090/s0273-0979-1989-15757-1 fatcat:4hucuixuffainnri2kus6bwpni

The Stability Theory of Belief

H. Leitgeb
2014 Philosophical Review  
The theory is based on three assumptions: the logical closure of rational belief; the axioms of probability for rational degrees of belief; and the so-called Lockean thesis, in which the concepts of rational  ...  of rational belief can be built around these principles that is not ad hoc but that has various philosophical features that are plausible independently.  ...  This is not just a random coincidence. From the principles of stability theory, one can derive such correspondence results for conditionalization and belief revision in general; see Leitgeb 2013a.  ... 
doi:10.1215/00318108-2400575 fatcat:rd4sy6mjwncnlhxi2pe77kt6j4

Remarks on generic stability in independent theories [article]

Gabriel Conant, Kyle Gannon
2019 arXiv   pre-print
In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of "generic stability" in arbitrary theories.  ...  Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure.  ...  In NIP theories, although not every type is necessarily definable and finitely satisfiable, the class of types with these properties is still quite resilient, and such types are now referred to as generically  ... 
arXiv:1905.11915v3 fatcat:ygeii5lisrcotkl4siqcs6wyl4

Strategic stability in Poisson games

Francesco De Sinopoli, Claudia Meroni, Carlos Pimienta
2014 Journal of Economic Theory  
Furthermore, we use such a space to define the corresponding strategically stable sets of equilibria.  ...  We show that they satisfy existence, admissibility, and robustness against iterated deletion of dominated strategies and inferior replies.  ...  Every Poisson game has a stable set.Existence of stable sets in Poisson games is a particular case of a more general existence result.  ... 
doi:10.1016/j.jet.2014.05.005 fatcat:rdtlgoqcfrbk7aee2cse5b25o4


2008 Advances in Algebra and Combinatorics  
A general result of model theory says that S has a model companion, denoted by T S , precisely when the class E of existentially closed S-sets is axiomatisable and in this case, T S axiomatises E.  ...  It is known that T S exists and is stable if and only if S is right coherent. In the study of stable first order theories, superstable and totally transcendental theories are of particular interest.  ...  It is not difficult to see that if p ∈ S(∅) (so that necessarily I p = ∅), then for any G-set A there is exactly one extension p A of p in S(A) with I p A = ∅.  ... 
doi:10.1142/9789812790019_0009 fatcat:wfs7os2rqjamzhsrra75qjhqr4

Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory

Saharon Shelah
1971 Annals of Mathematical Logic  
., some syntactical properties of uns able formulas, indiscernible sets and degrees of types in superstable theories.  ...  Introduction List of results, conjectures and pv blems connected to stability § 1. Notations § 2. Properties equivalent to unstabilit of formulas § 3.  ...  (See Def. 5.2.7 [See Shelah [D] Th. 3.1; IF] Th. 2.2, and here 5.8 are generalizations. ] There are some definitions of prime model; and proofs it exists under certain stability conditions.  ... 
doi:10.1016/0003-4843(71)90015-5 fatcat:q62p6hgaw5gzppxwzhp333mcye

Stabilization of port-Hamiltonian systems with discontinuous energy densities

Jochen Schmid
2022 Evolution Equations and Control Theory  
<p style='text-indent:20px;'>We establish an exponential stabilization result for linear port-Hamiltonian systems of first order with quite general, not necessarily continuous, energy densities.  ...  In particular, and in contrast to the previously known stabilization results, our result applies to vibrating strings or beams with jumps in their mass density and their modulus of elasticity.</p>  ...  I would like to thank the German Research Foundation (DFG) for financial support through the grant "Input-to-state stability and stabilization of distributed-parameter systems" (DA 767/7-1).  ... 
doi:10.3934/eect.2021063 fatcat:2hqle3pzpbeupavldxcbjfrd2q

Strong stochastic stability for non-uniformly expanding maps

2012 Ergodic Theory and Dynamical Systems  
A weaker form of stochastic stability was established in [AAr03] for a general class of non-uniformly expanding maps, in the sense of convergence of the physical measure to the SRB probability measure  ...  Consequently, we are able to obtain the strong stochastic stability for two examples of non-uniformly expanding maps.  ...  This does not necessarily imply that f is non-uniformly expanding.  ... 
doi:10.1017/s0143385712000077 fatcat:y57njtwnwncpfmafzoureyzklm

Numerical stability of Euclidean algorithm over ultrametric fields

Xavier Caruso
2017 Journal de Théorie des Nombres de Bordeaux  
The memory of a computer being necessarily finite, it is not possible to represent exhaustively all elements of W .  ...  We also refer to [6] for a complete exposition of the theory including many discussions and examples.  ...  Numerical stability of Euclidean algorithm over ultrametric fields To conclude with, let us comment on briefly the hypothesis (H).  ... 
doi:10.5802/jtnb.989 fatcat:ontuzlmyibgdngtmfzmegsdr2i

The axiom of choice and model-theoretic structures [article]

J K Truss
2019 arXiv   pre-print
We discuss the connections between the failure of the axiom of choice in set theory, and certain model-theoretic structures with enough symmetry.  ...  , but not necessarily conversely.  ...  The disadvantage is that they work in a weaker set theory, which we call FM, or possibly FMC (Fraenkel-Mostowski with choice) in which the axiom of extensionality is relaxed to allow the existence of '  ... 
arXiv:1908.11731v1 fatcat:bzfjzxdx7rh7tci7z4mqeqfa6i

On Zilber's field [article]

Ali Bleybel
2014 arXiv   pre-print
In this paper we use tools from set theory and the uncountable categoricity of Zilber's pseudo-exponential field to show that Zilber's field is isomorphic to the complex field with (standard) exponentiation  ...  and hence Schanuel's conjecture holds for that field.  ...  For later use, we will assume more generally that M is a countable transitive well founded model of ZFC. (So the conclusions of this section will hold for any such model).  ... 
arXiv:1310.3777v6 fatcat:vzhl4jhnsnbwpes6sfefngkqva

Borovik-Poizat rank and stability [article]

Jeffrey Burdges, Gregory Cherlin
2007 arXiv   pre-print
Poizat raised the issue of the relationship between this notion of rank and stability theory in the following terms: "un groupe de Borovik est une structure stable, alors qu'un univers rangé n'a aucune  ...  These axioms form the basis of the algebraic treatment of groups of finite Morley rank which is common today.  ...  So if the theory of M is unstable, we may suppose that there is a definable relation R on a definable subset X of M eq such that R linearly orders some infinite subset of X, not necessarily definable.  ... 
arXiv:0711.4040v1 fatcat:b5qku7m7gjampkw6remtof76vy

Page 3381 of Mathematical Reviews Vol. , Issue 88g [page]

1988 Mathematical Reviews  
Andreas Blass (1-MI) 88g:03086 03F35 03C45 Harnik, Victor (IL-HAIF) Set existence axioms for general (not necessarily countable) stability theory. Stability in model theory (Trento, 1984). Ann.  ...  In this paper the author develops an extension of Friedman’s ax- iom systems of “reverse mathematics” to study which set-theoretic existence axioms are necessary for model theory and algebra.  ... 
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