This paper has two parts. First, we establish some of the basic model theoretic facts about the T... more This paper has two parts. First, we establish some of the basic model theoretic facts about the Tsirelson space of Figiel and Johnson \cite{FJ}. Second, using the results of the first part, we give some facts about general Banach spaces. In particular, we show: (1) every {\em strongly separable} space, such as Tsirelson's space, has the {\em non independence property} (NIP); (2) every {\em $\omega$-categorical space} is stable if and only if it is strongly separable; consequently Tsirelson's space is not $\omega$-categorical; equivalently, its first-order theory (in any countable language) does not characterize Tsirelson's space, up to isometric isomorphism; (3) every {\em explicitly definable} space contains $c_0$ or $\ell_p$ ($1\leqslant p<\infty$), more precisely, if a Banach space $X$ is $\aleph_0$-categorical in any countable language, then at least one of the following two conditions holds: (a) there exists $1\leqslant p<\infty$ such that for every $\epsilon>0$ there exists a sequence $(x_n)$ which is $(1+\epsilon)$-equivalent to the standard unit basis of $\ell_p$; (b) for every $\epsilon>0$ there exists a sequence $(x_n)$ which is $(1+\epsilon)$-equivalent to the standard unit basis of $c_0$.
We study and characterize stability, NIP and NSOP in terms of topological and measure theoretical... more We study and characterize stability, NIP and NSOP in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, 'Talagrand's stability', and explain the relationship between this property and NIP in continuous logic. Using a result of Bourgain, Fremlin and Talagrand, we prove the 'almost definability' and 'Baire 1 definability' of coheirs assuming NIP. We show that a formula φ(x, y) has the strict order property if and only if there is a convergent sequence of continuous functions on the space of φ-types such that its limit is not continuous. We deduce from this a theorem of Shelah and point out the correspondence between this theorem and the Eberlein-Šmulian theorem.
We refine results of Gannon [G21, Theorem 4.7] and Simon [S15a, Lemma 2.8] on convergence of Morl... more We refine results of Gannon [G21, Theorem 4.7] and Simon [S15a, Lemma 2.8] on convergence of Morley sequences. We then introduce the notion of eventual N IP , as a property of a model, and prove a variant of [KP18, Corollary 2.2]. Finally, we give new characterizations of generically stable types (for countable theories) and reinforce the main result of Pillay [P18] on the model-theoretic meaning of Grothendieck's double limit theorem. * Partially supported by IPM grant 1400030118 1 [S15a, Lemma 2.8]. In this article, when we refer to Simon's Lemma, we mean this result. 2 Although we only use one analytical/combinatorial result (Fact 2.10), we will explain that it is not even needed and that all arguments in this paper are model-theoretic. Cf. Remark 2.12(iv) below. 3 We can consider countable fragments of (uncountable) theories, however, to make the proofs more readable, we assume that the theory is countable.
We generalize a theory of Shelah for continuous logic, namely a continuous theory has OP if and o... more We generalize a theory of Shelah for continuous logic, namely a continuous theory has OP if and only if it has IP or SOP.
We refine results of Gannon [G21, Theorem 4.7] and Simon [S15a, Lemma 2.8] on convergence of Morl... more We refine results of Gannon [G21, Theorem 4.7] and Simon [S15a, Lemma 2.8] on convergence of Morley sequences. We then introduce the notion of eventual N IP , as a property of a model, and prove a variant of [KP18, Corollary 2.2]. Finally, we give new characterizations of generically stable types (for countable theories) and reinforce the main result of [P18] on the model-theoretic meaning of Grothendieck's double limit theorem.
We generalize a theory of Shelah for continuous logic, namely a continuous theory has OP if and o... more We generalize a theory of Shelah for continuous logic, namely a continuous theory has OP if and only if it has IP or SOP.
In this short note, using results of Bourgain, Fremlin, and Talagrand [BFT78], we show that for a... more In this short note, using results of Bourgain, Fremlin, and Talagrand [BFT78], we show that for a countable structure M , a saturated elementary extension M * of M and a formula φ(x, y) the following are equivalent: (i) φ(x, y) is NIP on M (in the sense of Definition 2.1). (ii) Whenever p(x) ∈ S φ (M *) is finitely satisfiable in M then it is Baire 1 definable over M (in sense of Definition 2.5).
We give a new characterization of SOP (the strict order property) and an optimal version of a the... more We give a new characterization of SOP (the strict order property) and an optimal version of a theorem of Shelah, namely a theory has OP (the order property) if and only if it has IP (the independence property) or SOP. We point out some connections between dividing lines in first order theories and subclasses of Baire 1 functions.
We give several new characterizations of IP (the independence property) and SOP (the strict order... more We give several new characterizations of IP (the independence property) and SOP (the strict order property) for continuous first order logic and study their relations to the function theory and the Banach space theory. We suggest new dividing lines of unstable theories by the study of subclasses of Baire-1 functions and argue why one should not expect a perfect analog of Shelah's theorem, namely a theory is unstable iff it has IP or SOP , for real-valued logics, especially for continuous logic.
We study generically stable types/measures in both classical and continuous logics, and their con... more We study generically stable types/measures in both classical and continuous logics, and their connection with randomization and modes of convergence of types/measures. * Partially supported by IPM grant 1401030117. 1 These are abbreviations for definable and finitely satisfiable measure, finitely approximated measure, and frequency interpretation measure, respectively. Cf. [CGH21] for definitions.
This paper has three parts. First, we establish some of the basic model theoretic facts about $\m... more This paper has three parts. First, we establish some of the basic model theoretic facts about $\mathbb{T}$, the Tsirelson space of Figiel and Johnson \cite{FJ}. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In~particular, we show: (1) every \emph{strongly separable} space, such as $\mathbb{T}$, has the \emph{non independence property} (NIP); (2) every \emph{$\aleph_0$-categorical space} is stable if and only if it is strongly separable; consequently $\mathbb{T}$ is not $\aleph_0$-categorical; equivalently, its first-order theory (in any countable language) does not characterize $\mathbb{T}$, up to isometric isomorphism; (3) every \emph{explicitly definable} space contains isometric copies of $c_0$ or $\ell_p$ ($1\leqslant p<\infty$).
This paper has three parts. First, we establish some of the basic model theoretic facts about T, ... more This paper has three parts. First, we establish some of the basic model theoretic facts about T, the Tsirelson space of Figiel and Johnson FJ. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In particular, we show: (1) every strongly separable space, such as T, has the non independence property (NIP); (2) every _0-categorical space is stable if and only if it is strongly separable; consequently T is not _0-categorical; equivalently, its first-order theory (in any countable language) does not characterize T, up to isometric isomorphism; (3) every explicitly definable space contains isometric copies of c_0 or ℓ_p (1≤ p<∞).
A separable Banach space $X$ is said to be finitely determined if for each separable space $Y$ su... more A separable Banach space $X$ is said to be finitely determined if for each separable space $Y$ such that $X$ is finitely representable (f.r.) in $Y$ and $Y$ is f.r. in $X$ then $Y$ is isometric to $X$. We provide a direct proof (without model theory) of the fact that every finitely determined space $X$ (isometrically) contains every (separable) space $Y$ which is finitely representable in $X$. We also point out how a similar argument proves the Krivine-Maurey theorem on stable Banach spaces, and give the model theoretic interpretations of some results.
This paper has two parts. First, we complete the proof of the Kolmogorov extension theorem for un... more This paper has two parts. First, we complete the proof of the Kolmogorov extension theorem for unbounded random variables using compactness theorem of integral logic which was proved for bounded case in [8]. Second, we give a proof of the Eberlein-Smulian compactness theorem by Ramsey's theorem and point out the correspondence between this theorem and a result in Shelah's classification theory.
We introduce the notion of dependence, as a property of a Keisler measure, and generalize several... more We introduce the notion of dependence, as a property of a Keisler measure, and generalize several results of [HPS13] on generically stable measures (in NIP theories) to arbitrary theories. Among other things, we show that this notion is very natural and fundamental for several reasons: (i) all measures in NIP theories are dependent, (ii) all types and all fim measures in any theory are dependent, and (iii) as a crucial result in measure theory, the Glivenko-Cantelli class of functions (formulas) is characterized by dependent measures.
This paper has three parts. First, we establish some of the basic model theoretic facts about $\m... more This paper has three parts. First, we establish some of the basic model theoretic facts about $\mathbb{T}$, the Tsirelson space of Figiel and Johnson \cite{FJ}. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In~particular, we show: (1) every \emph{strongly separable} space, such as $\mathbb{T}$, has the \emph{non independence property} (NIP); (2) every \emph{$\aleph_0$-categorical space} is stable if and only if it is strongly separable; consequently $\mathbb{T}$ is not $\aleph_0$-categorical; equivalently, its first-order theory (in any countable language) does not characterize $\mathbb{T}$, up to isometric isomorphism; (3) every \emph{explicitly definable} space contains isometric copies of $c_0$ or $\ell_p$ ($1\leqslant p<\infty$).
In \cite{K3} we pointed out the correspondence between a result of Shelah in model theory, i.e. a... more In \cite{K3} we pointed out the correspondence between a result of Shelah in model theory, i.e. a theory is unstable if and only if it has IP or SOP, and the well known compactness theorem of Eberlein and \v{S}mulian in functional analysis. In this paper, we relate a {\em natural} Banach space $V$ to a formula $\phi(x,y)$, and show that $\phi$ is stable (resp NIP, NSOP) if and only if $V$ is reflexive (resp Rosenthal, weakly sequentially complete) Banach space. Also, we present a proof of the Eberlein-\v{S}mulian theorem by a model theoretic approach using Ramsey theorems which is illustrative to show some correspondences between model theory and Banach space theory.
A well-known theorem of Shelah asserts that a theory has $OP$ (the order property) if and only if... more A well-known theorem of Shelah asserts that a theory has $OP$ (the order property) if and only if it has $IP$ (the independence property) or $SOP$ (the strict order property). We give a mild strengthening of Shelah's theorem for classical logic and a generalization of his theorem for continuous logic.
E. Odell showed that Tsirelson's space is strongly separable. This result implies that Tsirel... more E. Odell showed that Tsirelson's space is strongly separable. This result implies that Tsirelson's space has the non independence property (NIP).
This paper has two parts. First, we establish some of the basic model theoretic facts about the T... more This paper has two parts. First, we establish some of the basic model theoretic facts about the Tsirelson space of Figiel and Johnson \cite{FJ}. Second, using the results of the first part, we give some facts about general Banach spaces. In particular, we show: (1) every {\em strongly separable} space, such as Tsirelson's space, has the {\em non independence property} (NIP); (2) every {\em $\omega$-categorical space} is stable if and only if it is strongly separable; consequently Tsirelson's space is not $\omega$-categorical; equivalently, its first-order theory (in any countable language) does not characterize Tsirelson's space, up to isometric isomorphism; (3) every {\em explicitly definable} space contains $c_0$ or $\ell_p$ ($1\leqslant p<\infty$), more precisely, if a Banach space $X$ is $\aleph_0$-categorical in any countable language, then at least one of the following two conditions holds: (a) there exists $1\leqslant p<\infty$ such that for every $\epsilon>0$ there exists a sequence $(x_n)$ which is $(1+\epsilon)$-equivalent to the standard unit basis of $\ell_p$; (b) for every $\epsilon>0$ there exists a sequence $(x_n)$ which is $(1+\epsilon)$-equivalent to the standard unit basis of $c_0$.
We study and characterize stability, NIP and NSOP in terms of topological and measure theoretical... more We study and characterize stability, NIP and NSOP in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, 'Talagrand's stability', and explain the relationship between this property and NIP in continuous logic. Using a result of Bourgain, Fremlin and Talagrand, we prove the 'almost definability' and 'Baire 1 definability' of coheirs assuming NIP. We show that a formula φ(x, y) has the strict order property if and only if there is a convergent sequence of continuous functions on the space of φ-types such that its limit is not continuous. We deduce from this a theorem of Shelah and point out the correspondence between this theorem and the Eberlein-Šmulian theorem.
We refine results of Gannon [G21, Theorem 4.7] and Simon [S15a, Lemma 2.8] on convergence of Morl... more We refine results of Gannon [G21, Theorem 4.7] and Simon [S15a, Lemma 2.8] on convergence of Morley sequences. We then introduce the notion of eventual N IP , as a property of a model, and prove a variant of [KP18, Corollary 2.2]. Finally, we give new characterizations of generically stable types (for countable theories) and reinforce the main result of Pillay [P18] on the model-theoretic meaning of Grothendieck's double limit theorem. * Partially supported by IPM grant 1400030118 1 [S15a, Lemma 2.8]. In this article, when we refer to Simon's Lemma, we mean this result. 2 Although we only use one analytical/combinatorial result (Fact 2.10), we will explain that it is not even needed and that all arguments in this paper are model-theoretic. Cf. Remark 2.12(iv) below. 3 We can consider countable fragments of (uncountable) theories, however, to make the proofs more readable, we assume that the theory is countable.
We generalize a theory of Shelah for continuous logic, namely a continuous theory has OP if and o... more We generalize a theory of Shelah for continuous logic, namely a continuous theory has OP if and only if it has IP or SOP.
We refine results of Gannon [G21, Theorem 4.7] and Simon [S15a, Lemma 2.8] on convergence of Morl... more We refine results of Gannon [G21, Theorem 4.7] and Simon [S15a, Lemma 2.8] on convergence of Morley sequences. We then introduce the notion of eventual N IP , as a property of a model, and prove a variant of [KP18, Corollary 2.2]. Finally, we give new characterizations of generically stable types (for countable theories) and reinforce the main result of [P18] on the model-theoretic meaning of Grothendieck's double limit theorem.
We generalize a theory of Shelah for continuous logic, namely a continuous theory has OP if and o... more We generalize a theory of Shelah for continuous logic, namely a continuous theory has OP if and only if it has IP or SOP.
In this short note, using results of Bourgain, Fremlin, and Talagrand [BFT78], we show that for a... more In this short note, using results of Bourgain, Fremlin, and Talagrand [BFT78], we show that for a countable structure M , a saturated elementary extension M * of M and a formula φ(x, y) the following are equivalent: (i) φ(x, y) is NIP on M (in the sense of Definition 2.1). (ii) Whenever p(x) ∈ S φ (M *) is finitely satisfiable in M then it is Baire 1 definable over M (in sense of Definition 2.5).
We give a new characterization of SOP (the strict order property) and an optimal version of a the... more We give a new characterization of SOP (the strict order property) and an optimal version of a theorem of Shelah, namely a theory has OP (the order property) if and only if it has IP (the independence property) or SOP. We point out some connections between dividing lines in first order theories and subclasses of Baire 1 functions.
We give several new characterizations of IP (the independence property) and SOP (the strict order... more We give several new characterizations of IP (the independence property) and SOP (the strict order property) for continuous first order logic and study their relations to the function theory and the Banach space theory. We suggest new dividing lines of unstable theories by the study of subclasses of Baire-1 functions and argue why one should not expect a perfect analog of Shelah's theorem, namely a theory is unstable iff it has IP or SOP , for real-valued logics, especially for continuous logic.
We study generically stable types/measures in both classical and continuous logics, and their con... more We study generically stable types/measures in both classical and continuous logics, and their connection with randomization and modes of convergence of types/measures. * Partially supported by IPM grant 1401030117. 1 These are abbreviations for definable and finitely satisfiable measure, finitely approximated measure, and frequency interpretation measure, respectively. Cf. [CGH21] for definitions.
This paper has three parts. First, we establish some of the basic model theoretic facts about $\m... more This paper has three parts. First, we establish some of the basic model theoretic facts about $\mathbb{T}$, the Tsirelson space of Figiel and Johnson \cite{FJ}. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In~particular, we show: (1) every \emph{strongly separable} space, such as $\mathbb{T}$, has the \emph{non independence property} (NIP); (2) every \emph{$\aleph_0$-categorical space} is stable if and only if it is strongly separable; consequently $\mathbb{T}$ is not $\aleph_0$-categorical; equivalently, its first-order theory (in any countable language) does not characterize $\mathbb{T}$, up to isometric isomorphism; (3) every \emph{explicitly definable} space contains isometric copies of $c_0$ or $\ell_p$ ($1\leqslant p<\infty$).
This paper has three parts. First, we establish some of the basic model theoretic facts about T, ... more This paper has three parts. First, we establish some of the basic model theoretic facts about T, the Tsirelson space of Figiel and Johnson FJ. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In particular, we show: (1) every strongly separable space, such as T, has the non independence property (NIP); (2) every _0-categorical space is stable if and only if it is strongly separable; consequently T is not _0-categorical; equivalently, its first-order theory (in any countable language) does not characterize T, up to isometric isomorphism; (3) every explicitly definable space contains isometric copies of c_0 or ℓ_p (1≤ p<∞).
A separable Banach space $X$ is said to be finitely determined if for each separable space $Y$ su... more A separable Banach space $X$ is said to be finitely determined if for each separable space $Y$ such that $X$ is finitely representable (f.r.) in $Y$ and $Y$ is f.r. in $X$ then $Y$ is isometric to $X$. We provide a direct proof (without model theory) of the fact that every finitely determined space $X$ (isometrically) contains every (separable) space $Y$ which is finitely representable in $X$. We also point out how a similar argument proves the Krivine-Maurey theorem on stable Banach spaces, and give the model theoretic interpretations of some results.
This paper has two parts. First, we complete the proof of the Kolmogorov extension theorem for un... more This paper has two parts. First, we complete the proof of the Kolmogorov extension theorem for unbounded random variables using compactness theorem of integral logic which was proved for bounded case in [8]. Second, we give a proof of the Eberlein-Smulian compactness theorem by Ramsey's theorem and point out the correspondence between this theorem and a result in Shelah's classification theory.
We introduce the notion of dependence, as a property of a Keisler measure, and generalize several... more We introduce the notion of dependence, as a property of a Keisler measure, and generalize several results of [HPS13] on generically stable measures (in NIP theories) to arbitrary theories. Among other things, we show that this notion is very natural and fundamental for several reasons: (i) all measures in NIP theories are dependent, (ii) all types and all fim measures in any theory are dependent, and (iii) as a crucial result in measure theory, the Glivenko-Cantelli class of functions (formulas) is characterized by dependent measures.
This paper has three parts. First, we establish some of the basic model theoretic facts about $\m... more This paper has three parts. First, we establish some of the basic model theoretic facts about $\mathbb{T}$, the Tsirelson space of Figiel and Johnson \cite{FJ}. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In~particular, we show: (1) every \emph{strongly separable} space, such as $\mathbb{T}$, has the \emph{non independence property} (NIP); (2) every \emph{$\aleph_0$-categorical space} is stable if and only if it is strongly separable; consequently $\mathbb{T}$ is not $\aleph_0$-categorical; equivalently, its first-order theory (in any countable language) does not characterize $\mathbb{T}$, up to isometric isomorphism; (3) every \emph{explicitly definable} space contains isometric copies of $c_0$ or $\ell_p$ ($1\leqslant p<\infty$).
In \cite{K3} we pointed out the correspondence between a result of Shelah in model theory, i.e. a... more In \cite{K3} we pointed out the correspondence between a result of Shelah in model theory, i.e. a theory is unstable if and only if it has IP or SOP, and the well known compactness theorem of Eberlein and \v{S}mulian in functional analysis. In this paper, we relate a {\em natural} Banach space $V$ to a formula $\phi(x,y)$, and show that $\phi$ is stable (resp NIP, NSOP) if and only if $V$ is reflexive (resp Rosenthal, weakly sequentially complete) Banach space. Also, we present a proof of the Eberlein-\v{S}mulian theorem by a model theoretic approach using Ramsey theorems which is illustrative to show some correspondences between model theory and Banach space theory.
A well-known theorem of Shelah asserts that a theory has $OP$ (the order property) if and only if... more A well-known theorem of Shelah asserts that a theory has $OP$ (the order property) if and only if it has $IP$ (the independence property) or $SOP$ (the strict order property). We give a mild strengthening of Shelah's theorem for classical logic and a generalization of his theorem for continuous logic.
E. Odell showed that Tsirelson's space is strongly separable. This result implies that Tsirel... more E. Odell showed that Tsirelson's space is strongly separable. This result implies that Tsirelson's space has the non independence property (NIP).
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