2. Approximation and classification in the **ergodic** **theory** of **countable** **groups** A (probability-)measure preserving action of a (discrete) countably infinite group Γ on (X, µ) is a homomorphism a : Γ → A (X, µ). The **set** of all measure preserving actions of Γ on (X, µ) naturally forms a Polish space A(Γ, X, µ) on which A acts continuously by coordinate-wise conjugation. The orbit A · a of a ∈ A(Γ, X, µ) is called its conjugacy class and two actions a and b from A(Γ, X, µ) with the same conjugacy class are said to be conju- gate. We say that b is weakly contained in a if it is in the closure of the conjugacy class of a, and we call a and b weakly equivalent if each weakly contains the other. If a ∈ A(Γ, X, µ) and b ∈ A(Γ, Y, ν) are actions with different underlying probabilities spaces then we say that b is weakly contained in a if it is a factor (i.e., quotient) of some c ∈ A(Γ, X, µ) that is weakly equivalent to a. The weak containment relation is reflexive and transitive, and weak equivalence is therefore an equivalence relation. Weak containment of measure pre- serving actions was introduced by Kechris in [Kec10] as an **ergodic** theoretic analogue of weak containment of unitary representations, and it has proven to be a remarkably robust notion that accurately captures an intuition that one measure preserving action asymptoti- cally approximates or simulates another. Ab´ert and Elek have recently defined a compact Polish topology on the **set** of weak equivalence classes in which many important invariants of weak equivalence become continuous functions [AE11], [TD12c]. A fundamental the- orem regarding weak containment is due to Ab´ert and Weiss and concerns the Bernoulli shift action of Γ which we now define.

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Let G be a **countable** discrete sofic group, ( X, µ ) a standard probability space and T : G y X a measurable G-action preserving µ. In [14], Lewis Bowen defined the sofic entropy of (X, µ, T) relative to a sofic approximation under the hypothesis that the action admits a finite generating partition. The definition was extended to general ( X, µ, T ) by Kerr and Li in [61] and Kerr gave a more elementary approach in [58]. In [17] Bowen showed that when G is amenable, sofic entropy relative to any sofic approximation agrees with the standard Kolmogorov-Sinai entropy. Despite some notable successes such as the proof in [14] that Bernoulli shifts with distinct base-entropies are nonisomorphic, many aspects of the **theory** of sofic entropy are still relatively undeveloped.

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Proof. Let B be the **set** of all subsets B of X so that there is a Polish space Z and a continuous bijection from Z to B. By Theorem 1.2 all open sets are in B. By Proposition 1.4 we need to show that B is closed under **countable** intersections and **countable** disjoint unions. By Example 2 (V) It is clear that B is closed under taking **countable** disjoint unions. In order to show that it is closed under **countable** interesctions let B i ∈ B, Z i be Polish and g i : Z i → B i be a continuous bijection for

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Abstract: - Molodtsov in 1999 initiated the concept of soft **set** **theory**. Based on the work of Molodtsov, Maji et al. in 2003 introduced some of the operations of soft sets and gave some of their basic properties. Ali et al. in 2009 further introduced more new operations on soft sets. The algebraic structures of soft sets were studied by Aktas and Cagman who introduced the notions of soft **groups** and their algebraic properties, while Acar et al studied soft semi rings and their basic properties. In this paper, we discuss the notions of soft rings, soft sub rings, soft ideal of a soft ring and idealistic soft rings and some of their algebraic properties with some illustrative examples.

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Constructive Zermelo-Fraenkel **Set** **Theory** has emerged as a standard reference **theory** that relates to constructive predicative mathematics as ZFC relates to classical Cantorian mathematics. The general topic of Constructive **Set** **Theory** originated in the seminal 1975 paper of John Myhill (cf. [16]), where a specific ax- iom system CST was introduced. Myhill developed constructive **set** **theory** with the aim of isolating the principles underlying Bishop’s conception of what sets and functions are. Moreover, he wanted “these principles to be such as to make the process of formalization completely trivial, as it is in the classical case” ([16], p. 347). Indeed, while he uses other primitives in his **set** **theory** CST besides the notion of **set**, it can be viewed as a subsystem of ZF. The advantage of this is that the ideas, conventions and practise of the **set** theoretical presentation of ordi- nary mathematics can be used in the **set** theoretical development of constructive mathematics, too. Constructive **Set** **Theory** provides a standard **set** theoretical framework for the development of constructive mathematics in the style of Errett Bishop and Douglas Bridges [6] and is one of several such frameworks for con- structive mathematics that have been considered. Aczel subsequently modified Myhill’s CST and the resulting **theory** was called Zermelo-Fraenkel **set** **theory**, CZF. A hallmark of this **theory** is that it possesses a type-theoretic interpre- tation (cf. [1, 2, 3]). Specifically, CZF has a scheme called Subset Collection Axiom (which is a generalization of Myhill’s Exponentiation Axiom) whose for- malization was directly inspired by the type-theoretic interpretation.

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E e e e e e be a **set** of parameters with respect to U, where each parameter e i i , 1, 2, ,5 stands for ‗expensive‘, ‗beautiful‘, ‗cheap‘, ‗modern‘, ‗wooden‘, respectively and A e e e 1 , 2 , 3 E . Suppose a soft **set** (F,A) describes the attractions of the houses, such that

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Munkres mentions that an overly casual approach to **set** **theory** can lead to logical paradoxes. For example, this happens if we try to consider the “**set** of all sets.” During the early part of the twentieth century mathematicians realized that problems with such things could be avoided by stipulating that sets cannot be “too large,” and effective safeguards to eliminate such difficulties were built into the formal axioms for **set** **theory**. One simple and reliable way of avoiding such problems with the informal approach to **set** **theory** in this course this is to assume that all con- structions take place in some extremely large **set** that is viewed as universal. This is consistent with the G¨odel-Bernays-von Neumann axiomatic approach to **set** **theory**, in which one handles the problem by considering two types of collections of objects: The CLASSES can be fairly arbitrary, but the SETS are constrained by a simple logical condition (specifically, they need to belong to some other class). Further information can be found in the online **set** **theory** notes mentioned above. — Viewing everything as contained in some very large **set** is a recommended option for this course if difficulties ever arise.

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This contradiction is coming from the dual nature of the particle viewed as a wave. In the first capacity it has only local dependence; in the second (wave) capacity it has a global dependence. The classical logic has diffi- culties in resolving this paradox. Changing the classical logic to logic makes the paradox apparent. Particle has the local property or zero dependence with other particles, media has total dependence so it is a global unique entity. Now, in **set** **theory**, any element is independent from the other so disjoint **set** has no elements in common. With this condition we have known that the true/false logic can be applied and **set** **theory** is the principal foun- dation. Now with conditional probability and dependence by copula the long distance dependence has an effect on any individual entity that now is not isolate but can have different types of dependence or synchronism (con- strain) whose effect is to change the probability of any particle. So particle with different degree of dependence can be represented by a new type of **set** as fuzzy **set** in which the boundary is not completely defined or where we cannot separate a **set** in its parts as in the evidence **theory**. In conclusion the Feynman paradox and Bell violation can be explained at a new level of complexity by many valued logics and new types of **set** **theory**.

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The presentation above is typical of **set** **theory** texts in two respects. First, the new concept in question, the closed unbounded sets (or the club sets, as they are commonly called), is described using a definition, not a syntactic representation. Only one logical symbol is used ( ) and even this symbol is not primitive in the ⊂ language of **set** **theory**. The rest of the text is in ordinary English and involves previously defined **set** **theory** concepts. While a syntactic representation for club sets could perhaps be provided by unpacking the meaning of the terms “regular”, “uncountable”, “cardinal”, “unbounded”, “limit points”, and so on (which would then involve unpacking the terms used to define those terms like ordinal, cofinal,

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Lstp(a/A) = Lstp(b/A), respectively. For the general **theory** of model **theory**, of the Lascar **groups**, and of rosy theories, we refer to [6], [11], and [2], respectively. For the homology **theory** in model **theory**, see [4, 3]. A particular case of the first homology group with respect to thorn-forking in rosy theories is studied in [7],[8]. The main difference of the first homology **groups** introduced in this section from those in the references is that the **groups** are defined with respect to a fixed independence notion in an arbitrary **theory** as follows, not necessarily thorn-forking/Shelah-forking in rosy/simple theories. However, as the reader will see, all the arguments from the rosy **theory** context can follow in the general context.

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FROK AXIOKATIZATIOJ TO GBIBRALIZATIOJ OF SBT THBORY SAXUBL FBIDRICH LOJDOJ SCHOOL OF BCOID ICS AID POLITICAL SCIBJCB THESIS SUB ITTED FOR THE DEGRBE OF PH D UIIVERSITY OF LOIDOJ JUIB 1987 1 ABSTRACT T[.]

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Nonstandard **set** theories typically postulate validity of the axioms of ZFC, including the Axiom of Choice, in the standard (or internal) universe. They also postulate some versions of Transfer, Standardization and Idealization. An easy argument shows that the last three principles, together with ZF alone, imply the Boolean Prime Ideal Theorem (BPI):

In chapter 4 we utilise our discovery o f the unique normal subgroup F N (x) associated with an irreducible character X o f a 7 t-separable group G, to construct a subset P ^ G ) Q Irr(G) such that the elem ents o f Px(G) when restricted to the K-classes o f G, they form a basis for the 71- class functions o f G, provided that 2 e n o r IGI is odd. In [IS 3] Isaacs proves that there exists a uniquely defined such subset o f Irr(G) which he calls B*(G). W e show that, under the hypothesis that 2 e n o r G is o f odd order, our **set** P*(G) ■ B*(G), thus giving an alternative way to construct Isaacs' B ^ G ) .

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(1) The partitions µ and σ both have **countable** many blocks. By Proposition 12 the intervals [0, µ] and [0, σ] are isomorphic if and only if the partitions µ and σ have the same 1-reduced type. Moreover, by Proposition 14 weights of all elements in these intervals are the continuum. Thus any isomorphism from [0, µ] to [0, σ] preserves weights. In this case the variants (Part(M), ∗ µ ) and (Part(M ), ∗ σ ) are isomorphic if and only

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In th is section computational problems on abelian **groups** represented with a **set** of de fin in g re la tion s and c lo se ly related problems are Investigated. In p a rtic u la r, the problem of computing the order and the canonical s tru ctu re o f a f i n i t e or an In f in it e abelian group 1s examined. Consequently the c lo s e ly related problems of computing the Hermite normal form and the Smith normal form of a non-singular integer m atrix are considered. Moreover an e ffe ctive way of solving systems of lin e a r Dlophantine equations is studied.

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University of Warwick institutional repository: http://go.warwick.ac.uk/wrap A Thesis Submitted for the Degree of PhD at the University of Warwick http://go.warwick.ac.uk/wrap/3493 This [r]

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Let be a finitely generated non-elementary Fuchsian group acting in the hyperbolic disk D . The Markov coding, originally introduced in [5] to encode limit points of as infinite words in a fixed **set** of generators, also gives a canonical shortest form for words in . The coding is defined relative to a fixed choice of fundamental domain R for , which we suppose is a finite sided convex polygon with vertices contained in D ∂ D , such that the interior angle at each vertex is strictly less than π . By a side of R we mean the closure in D of the geodesic arc joining a pair of adjacent vertices. We allow the infinite area case in which some adjacent vertices on ∂ D are joined by an arc contained in ∂ D ; we do not count these arcs as sides of R . We assume that the sides of R are paired; that is, for each side s of R there is a (unique) element e ∈ such that e ( s ) is also a side of R and such that R and e (R) are adjacent along e ( s ) . (Notice that this includes the possibility that e ( s ) = s, in which case e is elliptic of order 2 and the side s contains the fixed point of e in its interior. The condition that the vertex angle is strictly less than π excludes the possibility that the fixed point of e is counted as a vertex of R .)

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As a rough rule of thumb, until one feels at ease with nonstandard analysis, it is best to apply the familiar rules of internal mathematics freely to elements, but to be careful when working with sets of elements. (From a foundational point of view, everything in mathematics is a **set**. For example, a real number is an equivalence class of Cauchy sequences of rational numbers. Even a natural number is a **set**: the number 0 is the empty **set**, the number 1 is the **set** whose only element is 0, the number 2 is the **set** whose only elements are 0 and 1, etc. When I refer to “elements” or “objects” rather than to sets, only a psychological distinction is intended.)

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Definition 1.1 (RST, Definition 8.3; see also RST, Proposition 8.6, and Section 2 of this paper.) A **set** L is a level **set** if for all x, y ∈ L, V(x) = V(y) implies x = y . Level sets are finite, and the relation v defined on L by x v y ↔ V(x) ⊆ V(y) is a well-ordering. We always describe level sets in the increasing order by v; ie, if L = {z 0 , z 1 , . . . , z ` } is a level **set**, then V(z 0 ) ⊂ V(z 1 ) ⊂ . . . ⊂ V(z ` ).

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In a joint paper with O’Donovan and Lessmann [17] and a book manuscript in prepa- ration [18], we attempt to demonstrate that elementary analysis at a beginner’s level can be developed from a few very simple axioms that form a small fragment of FRIST (but transcend RIST in some important aspects). Several high school level calculus courses in Geneva have been successfully taught with this approach in Spring 2009. This paper is concerned with metamathematics of internal relative **set** theories. The experience with writing [18] showed that principles beyond those of FRIST are useful for some more advanced arguments in analysis. Here we present an extension of FRIST denoted GRIST, and prove that GRIST is a conservative extension of ZFC. Moreover, GRIST is complete over ZFC in a technical sense (Corollary 12.7) and universal among all theories of relative standardness satisfying some minimal axioms (Corollary 12.6). The interpretation of GRIST in ZFC that we use is the same as the one given in Section 6 of [13] for a version of FRIST. It is quite complicated, and the proof that GRIST is valid in it even more so. In contrast, the axioms of GRIST are fairly simple (see below). For these reasons, and unlike the usual practice in traditional nonstandard analysis, use of GRIST as a foundation for development of mathematics has to be based on its axioms rather than its models.

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