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Directed Sets and Topological Spaces Definable in O-Minimal Structures
Directed sets and topological spaces definable in o-minimal structures. Pablo And´ujarGuerrero∗ Margaret E. M. Thomas ∗ Erik Walsbergy 2010 Mathematics Subject Classification. 03C64 (Primary), 54A20, 54A05, 54D30 (Secondary). Key words. o-minimality, directed sets, definable topological spaces. Abstract We study directed sets definable in o-minimal structures, show- ing that in expansions of ordered fields these admit cofinal definable curves, as well as a suitable analogue in expansions of ordered groups, and furthermore that no analogue holds in full generality. We use the theory of tame pairs to extend the results in the field case to definable families of sets with the finite intersection property. We then apply our results to the study of definable topologies. We prove that all de- finable topological spaces display properties akin to first countability, and give several characterizations of a notion of definable compactness due to Peterzil and Steinhorn [PS99] generalized to this setting. 1 Introduction The study of objects definable in o-minimal structures is motivated by the notion that o-minimality provides a rich but \tame" setting for the theories of said objects. In this paper we study directed sets definable in o-minimal structures, focusing on expansions of groups and fields. By \directed set" we mean a preordered set in which every finite subset has a lower (if downward ∗Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, U.S.A. E-mail addresses: [email protected] (And´ujarGuer- rero), [email protected] (Thomas) yDepartment of Mathematics, Statistics, and Computer Science, Department of Math- ematics, University of California, Irvine, 340 Rowland Hall (Bldg.# 400), Irvine, CA 92697-3875, U.S.A. -
Even Ordinals and the Kunen Inconsistency∗
Even ordinals and the Kunen inconsistency∗ Gabriel Goldberg Evans Hall University Drive Berkeley, CA 94720 July 23, 2021 Abstract This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity phenomenon: assuming choiceless large cardinal axioms, the properties of the cumulative hierarchy turn out to alternate between even and odd ranks. The second part of the paper explores the structure of ultrafilters under choiceless large cardinal axioms, exploiting the fact that these axioms imply a weak form of the author's Ultrapower Axiom [1]. The third and final part of the paper examines the consistency strength of choiceless large cardinals, including a proof that assuming DC, the existence of an elementary embedding j : Vλ+3 ! Vλ+3 implies the consistency of ZFC + I0. embedding j : Vλ+3 ! Vλ+3 implies that every subset of Vλ+1 has a sharp. We show that the existence of an elementary embedding from Vλ+2 to Vλ+2 is equiconsistent with the existence of an elementary embedding from L(Vλ+2) to L(Vλ+2) with critical point below λ. We show that assuming DC, the existence of an elementary embedding j : Vλ+3 ! Vλ+3 implies the consistency of ZFC + I0. By a recent result of Schlutzenberg [2], an elementary embedding from Vλ+2 to Vλ+2 does not suffice. 1 Introduction Assuming the Axiom of Choice, the large cardinal hierarchy comes to an abrupt halt in the vicinity of an !-huge cardinal. -
Handout from Today's Lecture
MA532 Lecture Timothy Kohl Boston University April 23, 2020 Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 1 / 26 Cardinal Arithmetic Recall that one may define addition and multiplication of ordinals α = ot(A, A) β = ot(B, B ) α + β and α · β by constructing order relations on A ∪ B and B × A. For cardinal numbers the foundations are somewhat similar, but also somewhat simpler since one need not refer to orderings. Definition For sets A, B where |A| = α and |B| = β then α + β = |(A × {0}) ∪ (B × {1})|. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 2 / 26 The curious part of the definition is the two sets A × {0} and B × {1} which can be viewed as subsets of the direct product (A ∪ B) × {0, 1} which basically allows us to add |A| and |B|, in particular since, in the usual formula for the size of the union of two sets |A ∪ B| = |A| + |B| − |A ∩ B| which in this case is bypassed since, by construction, (A × {0}) ∩ (B × {1})= ∅ regardless of the nature of A ∩ B. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 3 / 26 Definition For sets A, B where |A| = α and |B| = β then α · β = |A × B|. One immediate consequence of these definitions is the following. Proposition If m, n are finite ordinals, then as cardinals one has |m| + |n| = |m + n|, (where the addition on the right is ordinal addition in ω) meaning that ordinal addition and cardinal addition agree. Proof. The simplest proof of this is to define a bijection f : (m × {0}) ∪ (n × {1}) → m + n by f (hr, 0i)= r for r ∈ m and f (hs, 1i)= m + s for s ∈ n. -
Cofinal Stable Logics
Guram Bezhanishvili Cofinal Stable Logics Nick Bezhanishvili Julia Ilin Abstract. We generalize the (∧, ∨)-canonical formulas to (∧, ∨)-canonical rules, and prove that each intuitionistic multi-conclusion consequence relation is axiomatizable by (∧, ∨)-canonical rules. This yields a convenient characterization of stable superintuition- istic logics. The (∧, ∨)-canonical formulas are analogues of the (∧, →)-canonical formulas, which are the algebraic counterpart of Zakharyaschev’s canonical formulas for superintu- itionistic logics (si-logics for short). Consequently, stable si-logics are analogues of subframe si-logics. We introduce cofinal stable intuitionistic multi-conclusion consequence relations and cofinal stable si-logics, thus answering the question of what the analogues of cofinal subframe logics should be. This is done by utilizing the (∧, ∨, ¬)-reduct of Heyting alge- bras. We prove that every cofinal stable si-logic has the finite model property, and that there are continuum many cofinal stable si-logics that are not stable. We conclude with several examples showing the similarities and differences between the classes of stable, cofinal stable, subframe, and cofinal subframe si-logics. Keywords: Intuitionistic logic, Intuitionistic multi-conclusion consequence relation, Axiomatization, Heyting algebra, Variety, Universal class. Mathematics Subject Classification: 03B55 · 06D20. 1. Introduction Superintuitionistic logics (si-logics for short) are propositional logics extend- ing the intuitionistic propositional calculus IPC. Consistent si-logics are also known as intermediate logics as they are exactly the logics situated between IPC and the classical propositional calculus CPC.BytheG¨odel translation, si-logics are closely related to modal logics extending S4.Ade- tailed account of si-logics and how they relate to modal logics can be found in [8]. -
COFINAL TYPES of ULTRAFILTERS 1. Introduction We Say That a Poset
COFINAL TYPES OF ULTRAFILTERS DILIP RAGHAVAN AND STEVO TODORCEVIC Abstract. We study Tukey types of ultrafilters on !, focusing on the question of when Tukey reducibility is equivalent to Rudin-Keisler reducibility. We give several conditions under which this equivalence holds. We show that there are only c many ultrafilters that are Tukey below any basically generated ultrafil- ter. The class of basically generated ultrafilters includes all known ultrafilters <! that are not Tukey above [!1] . We give a complete characterization of all ultrafilters that are Tukey below a selective. A counterexample showing that Tukey reducibility and RK reducibility can diverge within the class of P-points is also given. 1. Introduction We say that a poset hD; ≤i is directed if any two members of D have an upper bound in D. A set X ⊂ D is unbounded in D if it doesn't have an upper bound in D. A set X ⊂ D is said to be cofinal in D if 8y 2 D9x 2 X [y ≤ x]. Given directed sets D and E, a map f : D ! E is called a Tukey map if the image (under f) of every unbounded subset of D is unbounded in E. A map g : E ! D is called a convergent map if the image (under g) of every cofinal subset of E is cofinal in D. It is easy to see that there is a Tukey map f : D ! E iff there exists a convergent g : E ! D. When this situation obtains, we say that D is Tukey reducible to E, and we write D ≤T E. -
Directed Sets and Cofinal Types by Stevo Todorcevic
transactions of the american mathematical society Volume 290, Number 2, August 1985 DIRECTED SETS AND COFINAL TYPES BY STEVO TODORCEVIC Abstract. We show that 1, w, ax, u x ux and ["iF" are the only cofinal types of directed sets of size S,, but that there exist many cofinal types of directed sets of size continuum. A partially ordered set D is directed if every two elements of D have an upper bound in D. In this note we consider some basic problems concerning directed sets which have their origin in the theory of Moore-Smith convergence in topology [12, 3, 19, 9]. One such problem is to determine "all essential kind of directed sets" needed for defining the closure operator in a given class of spaces [3, p. 47]. Concerning this problem, the following important notion was introduced by J. Tukey [19]. Two directed sets D and E are cofinally similar if there is a partially ordered set C in which both can be embedded as cofinal subsets. He showed that this is an equivalence relation and that D and E are cofinally similar iff there is a convergent map from D into E and also a convergent map from E into D. The equivalence classes of this relation are called cofinal types. This concept has been extensively studied since then by various authors [4, 13, 7, 8]. Already, from the first introduc- tion of this concept, it has been known that 1, w, ccx, w X cox and [w1]<" represent different cofinal types of directed sets of size < Kls but no more than five such types were known. -
Dualities in Equivariant Kasparov Theory
DUALITIES IN EQUIVARIANT KASPAROV THEORY HEATH EMERSON AND RALF MEYER Abstract. We study several duality isomorphisms between equivariant bivari- ant K-theory groups, generalising Kasparov’s first and second Poincar´eduality isomorphisms. We use the first duality to define an equivariant generalisation of Lefschetz invariants of generalised self-maps. The second duality is related to the descrip- ∗ tion of bivariant Kasparov theory for commutative C -algebras by families of elliptic pseudodifferential operators. For many groupoids, both dualities ap- ply to a universal proper G-space. This is a basic requirement for the dual Dirac method and allows us to describe the Baum–Connes assembly map via localisation of categories. Contents 1. Introduction 1 2. Preliminaries on groupoid actions 7 3. EquivariantKasparovtheoryforgroupoids 11 4. The first duality 13 5. Bundles of compact spaces 29 6. The second duality 32 7. Duals for bundles of smooth manifolds 37 8. Conclusion and outlook 52 References 53 1. Introduction The K-homology of a smooth compact manifold M is naturally isomorphic to the (compactly supported) K-theory of its tangent bundle TM via the map that assigns to a K-theory class on TM an elliptic pseudodifferential operator with appropriate symbol. Dually, the K-theory of M is isomorphic to the (locally finite) K-homology of TM. Both statements have bivariant generalisations, which identify Kasparov’s group KK∗ C(M1), C(M2) for two smooth compact manifolds firstly arXiv:0711.0025v3 [math.KT] 24 Jul 2009 ∗ lf C with K (TM1 × M2), secondly with K∗ (TM2 × M1) := KK∗(C0(TM2 × M1), ). In this article, we consider substantial generalisations of these two duality isomor- phisms in which we replace smooth compact manifolds by more general spaces and work equivariantly with respect to actions of locally compact groups or groupoids. -
Coloring Linear Orders with Rado's Partial Order 1
COLORING LINEAR ORDERS WITH RADO'S PARTIAL ORDER RICCARDO CAMERLO AND ALBERTO MARCONE Abstract. Let ¹R be the preorder of embeddability between countable linear orders colored with elements of Rado's partial or- der (a standard example of a wqo which is not a bqo). We show that ¹R has fairly high complexity with respect to Borel reducibil- ity (e.g. if P is a Borel preorder then P ·B ¹R), although its exact classi¯cation remains open. 1. Introduction This note is a contribution to the study of the relations of embed- dability for countable colored linear orderings from the point of view of descriptive set theory. Fixing ! as the support of a countable linear or- dering, we can consider the space LO of all linear orderings on !. This is a standard Borel space (actually a Polish space, see [Kec95]). For our purposes, to color a linear order means assigning to each element of the support an element from a ¯xed countable set C. A colored linear ordering on ! is thus an element of LO £ C!, which is also a standard Borel space. If Q is a partial order on C we have a natural relation of embed- ! 0 0 dability ¹Q on LO £ C de¯ned by letting (v;') ¹Q (v ;' ) if and only if there exists g : ! ! ! such that: (1) 8a; b 2 ! (a v b () g(a) v0 g(b)); (2) 8a 2 !'(a) Q'0(g(a)). 1 Then ¹Q is a §1 preorder which is not Borel and it can be studied in the framework of Borel reducibility. -
Primes and Irreducibles in Truncation Integer Parts of Real Closed Fields
Primes and Irreducibles in Truncation Integer Parts of Real Closed Fields. ¤ D. Biljakovic, M. Kochetov and S. Kuhlmanny 04. 06. 2004 Abstract Berarducci (2000) studied irreducible elements of the ring k((G<0))©Z, which is an integer part of the power series ¯eld k((G)) where G is an ordered divisible abelian group and k is an ordered ¯eld. Pitteloud (2001) proved that some of the irreducible elements constructed by Berarducci are actually prime. Both authors mainly con- centrated on the case of archimedean G. In this paper, we study truncation integer parts of any (non-archimedean) real closed ¯eld and generalize results of Berarducci and Pitteloud. To this end, we study the canonical integer part Neg (F ) © Z of any truncation closed sub¯eld F of k((G)), where Neg (F ) := F \ k((G<0)), and work out in detail how the general case can be reduced to the case of archimedean G. In particular, we prove that k((G<0)) © Z has (co¯nally many) prime elements for any ordered divisible abelian group G. Addressing a question in the paper of Berarducci, we show that every truncation integer part of a non-archimedean expo- nential ¯eld has a co¯nal set of irreducible elements. Finally, we apply our results to two important classes of exponential ¯elds: exponential algebraic power series and exponential-logarithmic power series. 0 Introduction An integer part (IP for short) Z of an ordered ¯eld K is a discretely ordered subring, with 1 as the least positive element, and such that for every x 2 K, there is a z 2 Z such that z · x < z + 1. -
Well-Orderings, Ordinals and Well-Founded Relations. an Ancient
Well-orderings, ordinals and well-founded relations. An ancient principle of arithmetic, that if there is a non-negative integer with some property then there is a least such, is useful in two ways: as a source of proofs, which are then said to be \by induction" and as as a source of definitions, then said to be \by recursion." An early use of induction is in Euclid's proof that every integer > 2 is a product of primes, (where we take \prime" to mean \having no divisors other than itself and 1"): if some number is not, then let n¯ be the least counter-example. If not itself prime, it can be written as a product m1m2 of two strictly smaller numbers each > 2; but then each of those is a product of primes, by the minimality of n¯; putting those two products together expresses n¯ as a product of primes. Contradiction ! An example of definition by recursion: we set 0! = 1; (n + 1)! = n! × (n + 1): A function defined for all non-negative integers is thereby uniquely specified; in detail, we consider an attempt to be a function, defined on a finite initial segment of the non-negative integers, which agrees with the given definition as far as it goes; if some integer is not in the domain of any attempt, there will be a least such; it cannot be 0; if it is n + 1, the recursion equation tells us how to extend an attempt defined at n to one defined at n + 1. So no such failure exists; we check that if f and g are two attempts and both f(n) and g(n) are defined, then f(n) = g(n), by considering the least n where that might fail, and again reaching a contradiction; and so, there being no disagreement between any two attempts, the union of all attempts will be a well-defined function, which is familiar to us as the factorial function. -
Mathematical Logic
Arch. Math. Logic (2009) 48:825–835 DOI 10.1007/s00153-009-0156-0 Mathematical Logic More fine structural global square sequences Martin Zeman Received: 20 February 2009 / Published online: 13 October 2009 © The Author(s) 2009. This article is published with open access at Springerlink.com Abstract We extend the construction of a global square sequence in extender mod- els from Zeman [8] to a construction of coherent non-threadable sequences and give a characterization of stationary reflection at inaccessibles similar to Jensen’s charac- terization in L. Keywords Global quare sequence · Fine structure · Extender model · Weakly compact cardinal · Stationary reflection AMS Subject Classification 03E05 · 03E45 · 03E55 This note presents a fine structural construction of a so-called (κ, A) sequence for certain stationary subsets A of an inaccessible cardinal κ as well as a characterization of weakly compact cardinals in fine structural extender models in terms of station- ary reflection. These results extend analogous results of Jensen for the constructible universe that originate in [3] and are described in more detail in [1]. Although the characterization of weakly compact cardinals in an extender model turns out to be exactly the same as in L, the proof requires a significant amount of extra work. Also, the author believes that the proof presented in this paper is more straightforward than that described in [3] and [1]. The exposition in this paper is based on extender models with Jensen’s λ-indexing of extenders introduced in [4]; see [7] as a reference. The paper builds on previous work on fine structural square sequences in extender models, in particular on [5,6] and Research partially supported by NSF grant DMS-0500799. -
The Set of All Countable Ordinals: an Inquiry Into Its Construction, Properties, and a Proof Concerning Hereditary Subcompactness
W&M ScholarWorks Undergraduate Honors Theses Theses, Dissertations, & Master Projects 5-2009 The Set of All Countable Ordinals: An Inquiry into Its Construction, Properties, and a Proof Concerning Hereditary Subcompactness Jacob Hill College of William and Mary Follow this and additional works at: https://scholarworks.wm.edu/honorstheses Part of the Mathematics Commons Recommended Citation Hill, Jacob, "The Set of All Countable Ordinals: An Inquiry into Its Construction, Properties, and a Proof Concerning Hereditary Subcompactness" (2009). Undergraduate Honors Theses. Paper 255. https://scholarworks.wm.edu/honorstheses/255 This Honors Thesis is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Undergraduate Honors Theses by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. The Set of All Countable Ordinals: An Inquiry into Its Construction, Properties, and a Proof Concerning Hereditary Subcompactness A thesis submitted in partial fulfillment of the requirement for the degree of Bachelor of Science with Honors in Mathematics from the College of William and Mary in Virginia, by Jacob Hill Accepted for ____________________________ (Honors, High Honors, or Highest Honors) _______________________________________ Director, Professor David Lutzer _________________________________________ Professor Vladimir Bolotnikov _________________________________________ Professor George Rublein _________________________________________