### Citations

494 |
Set Theory. An Introduction to Independence Proofs
- Kunen
- 1980
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Citation Context ...T for PT,Σ. We shall identify condition p = (ψ, F, ε) in PT,Σ with the expression ψ(d̄p) < ε and use notations T + p and T ∪ {ψ(d̄p) < ε} interchangeably. A recap of the standard forcing terminology (=-=[27]-=-, [29]) is in order. Subset G of PT,Σ is a filter if every two elements of G have a common extension in G. A subset D of PT,Σ is dense if every q ∈ PT,Σ has an extension in D. It is dense below some p... |

284 |
Classification theory and the number of nonisomorphic models, volume 92
- Shelah
- 1990
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Citation Context ...h the set of types omissible in some model of T is Σ12-complete? Significant notions of the first-order model theory such as stability can be expressed in terms of cardinalities of sets of types (see =-=[30]-=-). This carries over to logic of metric structures (see [3], [17, §5]). Complexity of the set of not necessarily complete types over a given theory may give some information about theories in logic of... |

149 |
Classical descriptive set theory, volume 156 of Graduate Texts in Mathematics
- Kechris
- 1995
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Citation Context ...omplicated than omitting complete types. Our results confirm this intuition and show that the problem is essentially intractable. An excellent reference for notions from the Descriptive Set Theory is =-=[22]-=-; see also the beginning of §2. Theorem 1. (5) There is a complete theory T in a separable language such that the set of all types omissible over a model of T is Π11complete. (6) There is a theory T i... |

110 |
Model theory for metric structures, in Model theory with applications to algebra and analysis
- Yaacov, Berenstein, et al.
- 2008
(Show Context)
Citation Context ...useful remarks. The first author and the second author’s visits to Toronto were partially supported by NSERC. 1. Preliminaries We assume that the reader is acquainted with logic of metric structures (=-=[3]-=-, [22]). We strictly follow the outline of this logic given in [3]. In particular, all metric structures are required to have diameter 1 and all formulas are [0, 1]-valued. All function and predicate ... |

107 |
Set theory. On the structure of the real line
- Bartoszynski, Judah
- 1995
(Show Context)
Citation Context ...inted with logic of metric structures ([3], [20]). We strictly follow the outline of this logic given in [3]. In particular, all metric structures are required to have diameter 1 and all formulas are =-=[0, 1]-=--valued. All function and predicate symbols are equipped with a fixed modulus of uniform continuity. Every structure is a complete metric space in which interpretations of functional and relational sy... |

90 |
Set Theory: On the structure of the real
- Bartoszyński, Judah
- 1995
(Show Context)
Citation Context ...inted with logic of metric structures ([3], [22]). We strictly follow the outline of this logic given in [3]. In particular, all metric structures are required to have diameter 1 and all formulas are =-=[0, 1]-=--valued. All function and predicate symbols are equipped with a fixed modulus of uniform continuity. Every structure is a complete metric space in which interpretations of functional and relational sy... |

65 | Regularity properties in the classification program for separable amenable C∗-algebras
- Elliott, Toms
(Show Context)
Citation Context ...ribed first-order properties in these classes. Also, some of the deepest recent results on classification of C*-algebras have equivalent formulation in the language of (metric) first-order logic (see =-=[13]-=-, the introduction to [28], and [14] or [16]). We have a machine for construction of C*-algebras with properties prescribed by a given theory. These are generic algebras obtained by Henkin constructio... |

40 |
Building models by games
- Hodges
- 1985
(Show Context)
Citation Context ...omissible, but not jointly omissible, in its models. The Omitting Types Theorem is one of the most useful methods for constructing models of first-order theories with prescribed properties (see [26], =-=[23]-=-, or any general text on model theory). It implies, among other facts, the following. (1) If T is a theory in a countable language, then the set of all n-types realized in every model of T is Borel in... |

35 | Model theory of operator algebras II: Model theory, preprint
- Farah, Hart, et al.
- 1004
(Show Context)
Citation Context ...s of functional and relational symbols respect this modulus. It is a straightforward exercise to see that our results apply to the modification of this logic adapted to operator algebras presented in =-=[17]-=-. For a formula φ(x̄), structure of the same language M and a tuple ā in M by φ(ā)M we denote the interpretation of φ(x̄) at ā in M . In logic of metric structures there are two ways in which one c... |

30 |
Invariant descriptive set theory
- Gao
- 2008
(Show Context)
Citation Context ...natural continuous action of the permutation group S∞. This observation is a rich source of results on the interface between (classical first-order) model theory and descriptive set theory (see e.g., =-=[20]-=-). The space of separable metric structures of a fixed separable language L can be construed as a standard Borel space in more than one way. In [12] it was shown that every metric L-structure can be c... |

17 |
Invariant descriptive set theory, volume 293
- Gao
- 2009
(Show Context)
Citation Context ...natural continuous action of the permutation group S∞. This observation is a rich source of results on the interface between (classical first-order) model theory and descriptive set theory (see e.g., =-=[18]-=-). The space of metric structures of a fixed separable language L can be construed as a standard Borel space in more than one way. In [11] it was shown that every metric Lstructure can be canonically ... |

15 |
The descriptive set theory of C∗-algebra invariants
- Farah, Toms, et al.
(Show Context)
Citation Context ...d a function γR : ω n → [0, 1] corresponding to the interpretation of R in M(γ). The straightforward details are omitted. Space M̂(L) is similar to the space of separable C*-algebras Γ̂ introduced in =-=[19]-=-. Although M̂(L) is different from the Borel space of L-structures M(L) defined in [12], these two spaces are equivalent in the sense of [19]. The proof of this fact is analogous to the proof given in... |

13 | Model theory for metric structures
- Yaacov, Berenstein, et al.
(Show Context)
Citation Context ...e and Robin Tucker-Drob for very useful remarks, and to Bradd Hart for a large number of very useful remarks. 1. Preliminaries We assume that the reader is acquainted with logic of metric structures (=-=[3]-=-, [20]). We strictly follow the outline of this logic given in [3]. In particular, all metric structures are required to have diameter 1 and all formulas are [0, 1]-valued. All function and predicate ... |

10 |
Forcing and the omitting types theorem
- Keisler
- 1973
(Show Context)
Citation Context ...ately omissible, but not jointly omissible, in its models. The Omitting Types Theorem is one of the most useful methods for constructing models of first-order theories with prescribed properties (see =-=[26]-=-, [23], or any general text on model theory). It implies, among other facts, the following. (1) If T is a theory in a countable language, then the set of all n-types realized in every model of T is Bo... |

8 | A proof of completeness for continuous first-order logic
- Yaacov, Pedersen
(Show Context)
Citation Context ...gly set Th(M) = {φ : φM = 0}. Alternatively, one may consider the functional φ 7→ φM as the theory of M . We shall tacitly use completeness theorem for logic of metric structures whenever convenient (=-=[6]-=-). 1.1. Conditions and types. A closed condition is an expression of the form φ(x̄) = 0 for a formula φ(x̄) and type is a set of conditions. (Open conditions will be defined in §1.1.1 below.) It is re... |

8 |
Set theory: exploring independence and truth
- Schindler
(Show Context)
Citation Context ...PT,Σ. We shall identify condition p = (ψ, F, ε) in PT,Σ with the expression ψ(d̄p) < ε and use notations T + p and T ∪ {ψ(d̄p) < ε} interchangeably. A recap of the standard forcing terminology ([27], =-=[29]-=-) is in order. Subset G of PT,Σ is a filter if every two elements of G have a common extension in G. A subset D of PT,Σ is dense if every q ∈ PT,Σ has an extension in D. It is dense below some p ∈ PT,... |

7 | Omitting types and AF algebras
- Carlson, Cheung, et al.
- 2014
(Show Context)
Citation Context ...because some of the most important properties of C*-algebras are uniformly definable by a sequence of types. This includes nuclearity, nuclear dimension, decomposition rank, and being TAF, AF or UHF (=-=[10]-=-, [16]). These types are 26 ILIJAS FARAH AND MENACHEM MAGIDOR particularly simple and we include a straightforward technical sharpening of Proposition 4.4 with an eye to potential applications. A unif... |

7 | The isomorphism relation for separable C*-algebras
- Elliott, Farah, et al.
- 2012
(Show Context)
Citation Context ... model theory and descriptive set theory (see e.g., [20]). The space of separable metric structures of a fixed separable language L can be construed as a standard Borel space in more than one way. In =-=[12]-=- it was shown that every metric L-structure can be canonically extended to one whose universe is the Urysohn metric space. Borel space M(L) of all L-structures obtained in this way is not convenient f... |

6 |
Classical descriptive set theory. Vol. 156. Graduate Texts in Mathematics
- Kechris
- 1995
(Show Context)
Citation Context ...thus show that (1) fails and confirm the intuition that the problem is essentially intractable. Our results are expressed in the language of descriptive set theory, an excellent reference to which is =-=[24]-=-; see also the beginning of §2. Theorem 1. (5) There is a complete theory T in a separable language such that the set of all types omissible over a model of T is Π11complete. (6) There is a theory T i... |

5 | Logic and operator algebras
- Farah
- 2014
(Show Context)
Citation Context ...N , then c ∈ ωω. Therefore the tree Rc is ill-founded, and tR is realized. 3. Forcing and omitting types Our study of generic models is motivated by potential applications to operator algebras (see =-=[14]-=-, [15], §4 and §6). Results related to our results were obtained in [4] and [11], similarly inspired by Keisler’s classic [26]. Both of these papers study a version of Keisler’s forcing adapted to the... |

5 |
Continuous model theory and its applications, 2012, Course notes, available at http://www.math.mcmaster.ca/∼bradd/courses/math712/index.html
- Hart
(Show Context)
Citation Context ...l remarks. The first author and the second author’s visits to Toronto were partially supported by NSERC. 1. Preliminaries We assume that the reader is acquainted with logic of metric structures ([3], =-=[22]-=-). We strictly follow the outline of this logic given in [3]. In particular, all metric structures are required to have diameter 1 and all formulas are [0, 1]-valued. All function and predicate symbol... |

4 | Model theoretic forcing in analysis
- Yaacov, Iovino
(Show Context)
Citation Context ...d. 3. Forcing and omitting types Our study of generic models is motivated by potential applications to operator algebras (see [14], [15], §4 and §6). Results related to our results were obtained in =-=[4]-=- and [11], similarly inspired by Keisler’s classic [26]. Both of these papers study a version of Keisler’s forcing adapted to the infinitary version of the logic of metric structures. In the classical... |

4 | Operator Algebras, vol. 122, Encyclopaedia of Mathematical Sciences. Subseries: Operator Algebras and Non-commutative Geometry, no - Blackadar - 2006 |

4 | Rigidity of continuous quotients - Farah, Shelah - 2014 |

4 |
The descriptive classification of some classes of C∗-algebras
- Kechris
- 1996
(Show Context)
Citation Context ...ass is known. Results of [16] (combined with §3 and §4) reduce several prominent open problems on classification of C*-algebras to problems about the existence of theories with certain properties. In =-=[25]-=- Kechris defined a Borel space of C*-algebras and proved that the nuclear C*-algebras form its Borel subset. We give a generalization of this result. Recall that a Borel structure on the space of mode... |

3 | A Lopez-Escobar theorem for continuous logic, preprint. arXiv 1407.7102 - Yaacov, Nies, et al. |

3 |
Existentially closed II1 factors. arXiv preprint arXiv:1310.5138
- Farah, Goldbring, et al.
- 2013
(Show Context)
Citation Context ...en c ∈ ωω. Therefore the tree Rc is ill-founded, and tR is realized. 3. Forcing and omitting types Our study of generic models is motivated by potential applications to operator algebras (see [14], =-=[15]-=-, §4 and §6). Results related to our results were obtained in [4] and [11], similarly inspired by Keisler’s classic [26]. Both of these papers study a version of Keisler’s forcing adapted to the infin... |

3 |
Model theory of nuclear C*-algebras
- Farah, Hart, et al.
(Show Context)
Citation Context ...bras form the largest reasonably well-behaved and well-studied class of C*-algebras). C*-algebras which are UHF, AF ([10]), nuclear, of finite nuclear dimension, of finite decomposition rank, or TAF (=-=[16]-=-) are uniformly definable by a sequence of universal types. Therefore Theorem 4.2, Theorem 4.3, Proposition 4.4, Proposition 4.5 and Corolary 4.6 open possibilities for constructing C*-algebras with p... |

3 | On Kirchberg’s embedding problem
- Goldbring, Sinclair
(Show Context)
Citation Context ...M is isomorphic to a submodel of N , ā ∈M , and φ(x̄, ȳ) is a quantifier-free L-formula then inf ȳ φ(ā, ȳ)M = inf ȳ φ(ā, ȳ)N . Existentially closed C*-algebras and II1 factors were studied in =-=[21]-=- and [15], respectively. Corollary 4.6. Assume T is an ∀∃-axiomatizable theory in a separable language and tn, for n ∈ ω, is a uniform sequence of universal types. If the class M of all models of T th... |

3 |
Nuclear dimension and
- Sato, White, et al.
(Show Context)
Citation Context ...es in these classes. Also, some of the deepest recent results on classification of C*-algebras have equivalent formulation in the language of (metric) first-order logic (see [13], the introduction to =-=[28]-=-, and [14] or [16]). We have a machine for construction of C*-algebras with properties prescribed by a given theory. These are generic algebras obtained by Henkin construction as described in §3 and §... |

2 |
A brief note on omitting partial types in continuous model theory
- Bice
- 2012
(Show Context)
Citation Context ...then any sequence tn, for n ∈ ω, of complete types each of which can be omitted in a model of T can be simultaneously omitted in a model of T. Examples constructed by I. Ben-Yaacov ([2]) and T. Bice (=-=[7]-=-) demonstrate that omitting partial types in logic of metric structures is inherently more complicated than omitting complete types. We find a high lower bound for the descriptive complexity of the se... |

2 | Omitting uncountable types, and the strength of [0, 1]- valued logics
- Caicedo, Iovino
(Show Context)
Citation Context ... for complete types (by [3] or Proposition 3.1) but not for partial types (see Corollary 5.4). There are several good sources for model-theoretic forcing in the context of logic of metric structures (=-=[9]-=-, [11], [4], [21, Appendix A]). Since the present paper is a companion to [16] meant to be self-contained and accessible to non-logicians, we include some of the basics for the reader’s convenience. T... |

2 | Omitting types for infinitary [0, 1]-valued logic
- Eagle
(Show Context)
Citation Context ...Forcing and omitting types Our study of generic models is motivated by potential applications to operator algebras (see [14], [15], §4 and §6). Results related to our results were obtained in [4] and =-=[11]-=-, similarly inspired by Keisler’s classic [26]. Both of these papers study a version of Keisler’s forcing adapted to the infinitary version of the logic of metric structures. In the classical situatio... |

2 |
On Kirchberg’s embedding problem. arXiv:1404.1861
- Goldbring, Sinclair
- 2014
(Show Context)
Citation Context ...M is isomorphic to a submodel of N , ā ∈M , and φ(x̄, ȳ) is a quantifier-free L-formula then inf ȳ φ(ā, ȳ)M = inf ȳ φ(ā, ȳ)N . Existentially closed C*-algebras and II1 factors were studied in =-=[19]-=- and [13], respectively. Corollary 4.7. Assume T is an ∀∃-axiomatizable theory in a separable language and tn, for n ∈ ω, is a uniform sequence of universal types. If the class M of all models of T th... |

1 | Definability of groups in ℵ0-stable metric structures - Yaacov |