13 research outputs found

    Classification from a computable viewpoint

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    Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we describe some recent work on classification in computable structure theory. Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings. Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work

    On what I do not understand (and have something to say): Part I

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    This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (``see ... '' means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The other half, concentrating on model theory, will subsequently appear

    Complementation and decompositions in some weakly Lindelöf Banach spaces

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    AbstractLet Γ denote an uncountable set. We consider the questions if a Banach space X of the form C(K) of a given class (1) has a complemented copy of c0(Γ) or (2) for every c0(Γ)⊆X has a complemented c0(E) for an uncountable E⊆Γ or (3) has a decomposition X=A⊕B where both A and B are nonseparable. The results concern a superclass of the class of nonmetrizable Eberlein compacts, namely Ks such that C(K) is Lindelöf in the weak topology and we restrict our attention to Ks scattered of countable height. We show that the answers to all these questions for these C(K)s depend on additional combinatorial axioms which are independent of ZFC±CH. If we assume the P-ideal dichotomy, for every c0(Γ)⊆C(K) there is a complemented c0(E) for an uncountable E⊆Γ, which yields the positive answer to the remaining questions. If we assume ♣, then we construct a nonseparable weakly Lindelöf C(K) for K of height ω+1 where every operator is of the form cI+S for c∈R and S with separable range and conclude from this that there are no decompositions as above which yields the negative answer to all the above questions. Since, in the case of a scattered compact K, the weak topology on C(K) and the pointwise convergence topology coincide on bounded sets, and so the Lindelöf properties of these two topologies are equivalent, many results concern also the space Cp(K)

    Set Theory

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    This stimulating workshop exposed some of the most exciting recent develops in set theory, including major new results about the proper forcing axiom, stationary reflection, gaps in P(ω)/Fin, iterated forcing, the tree property, ideals and colouring numbers, as well as important new applications of set theory to C*-algebras, Ramsey theory, measure theory, representation theory, group theory and Banach spaces

    Combinatorics of countable ordinal topologies

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    We study combinatorial properties of ordinals under the order topology, focusing on the subspaces, partition properties and autohomeomorphism groups of countable ordinals. Our main results concern topological partition relations. Let n be a positive integer, let κ be a cardinal, and write [X] n for the set of subsets of X of size n. Given an ordinal β and ordinals αi for all i ∈ κ, write β →top (αi) n i∈κ to mean that for every function c : [β] n → κ (a colouring) there is some subspace X ⊆ β and some i ∈ κ such that X is homeomorphic to αi and [X] n ⊆ c −1 ({i}). We examine the cases n = 1 and n = 2, defining the topological pigeonhole number P top (αi) i∈κ to be the least ordinal β (when one exists) such that β →top (αi) 1 i∈κ , and the topological Ramsey number Rtop (αi) i∈κ to be the least ordinal β (when one exists) such that β →top (αi) 2 i∈κ . We resolve the case n = 1 by determining the topological pigeonhole number of an arbitrary sequence of ordinals, including an independence result for one class of cases. In the case n = 2, we prove a topological version of the Erd˝os–Milner theorem, namely that Rtop (α, k) is countable whenever α is countable and k is finite. More precisely, we prove that Rtop(ω ω β , k + 1) ≤ ω ω β·k for all countable ordinals β and all positive integers k. We also provide more careful upper bounds for certain small ordinals, including Rtop(ω + 1, k + 1) = ω k + 1, Rtop(α, k) < ωω whenever α < ω2 , Rtop(ω 2 , k) ≤ ω ω and Rtop(ω 2 + 1, k + 2) ≤ ω ω·k + 1 for all positive integers k. Outside the partition calculus, we prove a topological analogue of Hausdorff’s theorem on scattered total orderings. This allows us to characterise countable subspaces of ordinals as the order topologies of countable scattered total orderings. As an application, we compute the number of subspaces of an ordinal up to homeomorphism. Finally, we study the group of autohomeomorphisms of ω n ·m+1 for finite n and m. We classify the normal subgroups contained in the pointwise stabiliser of the limit points. These subgroups fall naturally into D (n) disjoint sets, each either countable or of size 22 ℵ0 , where D (n) is the number of ⊆-antichains of P ({1, 2, . . . , n}). Our techniques span a variety of disciplines, including set theory, general topology and permutation group theor
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