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Guessing Axioms, Invariance and Suslin Trees
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of East Anglia digital repository Guessing Axioms, Invariance and Suslin Trees A thesis submitted to the School of Mathematics of the University of East Anglia in partial fulfilment of the requirements for the degree of Doctor of Philosophy By Alexander Primavesi November 2011 c This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that its copyright rests with the author and that use of any information derived there from must be in accordance with current UK Copyright Law. In addition, any quotation or extract must include full attribution. Abstract In this thesis we investigate the properties of a group of axioms known as `Guessing Axioms,' which can be used to extend the standard axiomatisation of set theory, ZFC. In particular, we focus on the axioms called `diamond' and `club,' and ask to what extent properties of the former hold of the latter. A question of I. Juhasz, of whether club implies the existence of a Suslin tree, remains unanswered at the time of writing and motivates a large part of our in- vestigation into diamond and club. We give a positive partial answer to Juhasz's question by defining the principle Superclub and proving that it implies the exis- tence of a Suslin tree, and that it is weaker than diamond and stronger than club (though these implications are not necessarily strict). Conversely, we specify some conditions that a forcing would have to meet if it were to be used to provide a negative answer, or partial answer, to Juhasz's question, and prove several results related to this. -
A New Order Theory of Set Systems and Better Quasi-Orderings
03 Progress in Informatics, No. 9, pp.9–18, (2012) 9 Special issue: Theoretical computer science and discrete mathematics Research Paper A new order theory of set systems and better quasi-orderings Yohji AKAMA1 1Mathematical Institute, Tohoku University ABSTRACT By reformulating a learning process of a set system L as a game between Teacher (presenter of data) and Learner (updater of abstract independent set), we define the order type dim L of L to be the order type of the game tree. The theory of this new order type and continuous, monotone function between set systems corresponds to the theory of well quasi-orderings (WQOs). As Nash-Williams developed the theory of WQOs to the theory of better quasi- orderings (BQOs), we introduce a set system that has order type and corresponds to a BQO. We prove that the class of set systems corresponding to BQOs is closed by any monotone function. In (Shinohara and Arimura. “Inductive inference of unbounded unions of pattern languages from positive data.” Theoretical Computer Science, pp. 191–209, 2000), for any set system L, they considered the class of arbitrary (finite) unions of members of L.Fromview- point of WQOs and BQOs, we characterize the set systems L such that the class of arbitrary (finite) unions of members of L has order type. The characterization shows that the order structure of the set system L with respect to the set inclusion is not important for the result- ing set system having order type. We point out continuous, monotone function of set systems is similar to positive reduction to Jockusch-Owings’ weakly semirecursive sets. -
Normal Tree Orders for Infinite Graphs
Normal tree orders for infinite graphs J.-M. Brochet and R. Diestel A well-founded tree T defined on the vertex set of a graph G is called normal if the endvertices of any edge of G are comparable in T . We study how normal trees can be used to describe the structure of infinite graphs. In particular, we extend Jung’s classical existence theorem for trees of height ω to trees of arbitrary height. Applications include a structure theorem for graphs without large complete topological minors. A number of open problems are suggested. 1. Introduction: normal spanning trees The aim of this paper is to see how a classical and powerful structural device for the study of countable graphs, the notion of a normal spanning tree, can be made available more generally. The existence of such spanning trees, while trivial in the finite case (where they are better known as depth-first search trees), is in general limited to countable graphs. By generalizing the graph theoretical trees involved to order theoretical trees, a concept better suited to express uncountably long ‘ends’, we shall be able to extend the classical existence theorems for normal trees to arbitrary cardinalities, while retaining much of their original strength. Throughout the paper, G will denote an arbitrary connected graph. Con- sider a tree T ⊆ G, with a root r, say. If T spans G, the choice of r imposes a partial order on the vertex set V (G) of G: write x 6 y if x lies on the unique r–y path in T . -
BETTER QUASI-ORDERS for UNCOUNTABLE CARDINALS T
Sh:110 ISRAEL JOURNAL OF MATHEMATICS, Vol. 42, No. 3, 1982 BETTER QUASI-ORDERS FOR UNCOUNTABLE CARDINALS t BY SAHARON SHELAH ABSTRACT We generalize the theory of Nash-Williams on well quasi-orders and better quasi-orders and later results to uncountable cardinals. We find that the first cardinal K for which some natural quasi-orders are K-well-ordered, is a (specific) mild large cardinal. Such quasi-orders are ~ (the class of orders which are the union of <- A scattered orders) ordered by embeddability and the (graph theoretic) trees under embeddings taking edges to edges (rather than to passes). CONTENTS §0. Introduction ................................................................................................................. 178 ]Notation, including basic definitions and claims on barriers.] §1. The basic generalization ............................................................................................... 186 [We define K-X-bqo, and prove it is preserved by ~a (Q).] §2. Existence theorem and a stronger notion suitable for powers ....................................... 193 [We define the relevant large cardinals, and prove the existence of non-trivially K-X-bqos, prove that the narrowness cardinal of a Q is some large cardinal and introduce e.g. "[K]-X-bqo" which is preserved by products.] §3. Examples ...................................................................................................................... 201 §4. More information ........................................................................................................ -
AUTOMORPHISMS of A^-TREES
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 173, November 1972 AUTOMORPHISMSOF a^-TREES BY THOMASJ. JECH(') ABSTRACT. The number of automorphisms of a normal c<Ji-tree T, denoted by o-(T), is either finite or 2 °<cr(T)¿ 2 . Moreover, if cr(T) is infinite then cr(T)^0 _ cr(T). Moreover, if T has no Suslin subtree then cr(T) is finite or v(T) - 2 " or cr(T) -2 '.It is consistent that there is a Suslin tree with arbitrary pre- cribed °~(T) between 2 u and 2 *, subject to the restriction above; e.g. 2 0 K, 2 1 = ^324 an(^ a(T) = N]7. We prove related results for Kurepa trees and isomorphism types of trees. We use Cohen's method of forcing and Jensen's tech- niques in L. 0. Introduction. A normal co.-tree, being a partially ordered set with K. elements, admits at most 2 automorphisms. Also, the number of isomorphism types of co .-trees is not larger that 2 . In §1, we shall show that the number of automorphisms a normal co .-tree admits is either finite, or at least 2 ; as a matter of fact, it is 2 or 2 , with the exception of Suslin trees. In §2, we investigate Suslin trees. Inspired by Jensen's result [6l, which says that it is consistent to have rigid Suslin trees, and Suslin trees which admit N ., or N~ automorphisms, we prove a general consistency result saying that the number of automorphisms of a Suslin tree can be anything between 2 and 2 it is allowed to be. -
On the Continuous Gradability of the Cut-Point Orders of $\Mathbb R $-Trees
On the continuous gradability of the cut-point orders of R-trees Sam Adam-Day* *Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom; [email protected] Abstract An R-tree is a certain kind of metric space tree in which every point can be branching. Favre and Jonsson posed the following problem in 2004: can the class of orders underlying R-trees be characterised by the fact that every branch is order-isomorphic to a real interval? In the first part, I answer this question in the negative: there is a ‘branchwise-real tree order’ which is not ‘continuously gradable’. In the second part, I show that a branchwise-real tree order is con- tinuously gradable if and only if every well-stratified subtree is R-gradable. This link with set theory is put to work in the third part answering refinements of the main question, yielding several independence results. For example, when κ ¾ c, there is a branchwise-real tree order which is not continuously gradable, and which satisfies a property corresponding to κ-separability. Conversely, under Martin’s Axiom at κ such a tree does not exist. 1 Introduction An R-tree is to the real numbers what a graph-theoretic tree is to the integers. Formally, let 〈X , d〉 be a metric space. An arc between x, y ∈ X is the image of a topological embedding r : [a, b] → X of a real interval such that r(a) = x and r(b) = y (allowing for the possibility that a = b). -
Parametrized ♢ Principles
Parametrized ♦ principles Justin Tatch Moore, Michael Hruˇs´ak, Mirna Dˇzamonja ∗† September 2, 2003 Abstract We will present a collection of guessing principles which have a similar relationship to ♦ as cardinal invariants of the continuum have to CH. The purpose is to provide a means for systematically analyzing ♦ and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of CH and ♦ in models such as those of Laver, Miller, and Sacks. 1 Introduction Very early on in the course of modern set theory, Jensen isolated the following combinatorial principle known as ♦: ♦ There is a sequence Aα (α < ω1) such that for all α < ω1, Aα ⊆ α and if X is a subset of ω1 then set {α < ω1 : X ∩ α = Aα} is stationary. ∗The first and third authors received support from EPSRC grant GR/M71121 for the research of this paper. The research of the second author was supported in part by the Netherlands Organization for Scientific Research (NWO) – Grant 613.007.039, and in part by the Grant Agency of the Czech Republic – Grant GACRˇ 201/00/1466. †2000 Mathematics Subject Classification. Primary 03E17, 03E65. Key words: Dia- mond, weak diamond, cardinal invariant, guessing principle. 1 Jensen used this principle to construct a Suslin tree [17] and later many other constructions were carried out using ♦ as an assumption — see [9]. The purpose of this paper is to provide a broad framework for analyzing the consequences of Jensen’s ♦ principle. Our intent is to present an array of ♦-principles which have the same relation to ♦ as the cardinal invariants of the continuum (see e.g. -
1. the Zermelo Fraenkel Axioms of Set Theory
1. The Zermelo Fraenkel Axioms of Set Theory The naive definition of a set as a collection of objects is unsatisfactory: The objects within a set may themselves be sets, whose elements are also sets, etc. This leads to an infinite regression. Should this be allowed? If the answer is “yes”, then such a set certainly would not meet our intuitive expectations of a set. In particular, a set for which A A holds contradicts our intuition about a set. This could be formulated as a first∈ axiom towards a formal approach towards sets. It will be a later, and not very essential axiom. Not essential because A A will not lead to a logical contradiction. This is different from Russel’s paradox:∈ Let us assume that there is something like a universe of all sets. Given the property A/ A it should define a set R of all sets for which this property holds. Thus R = A∈ A/ A . Is R such a set A? Of course, we must be able to ask this { | ∈ } question. Notice, if R is one of the A0s it must satisfy the condition R/ R. But by the very definition of R, namely R = A A/ A we get R R iff R/∈ R. Of course, this is a logical contradiction. But{ how| can∈ } we resolve it?∈ The answer∈ will be given by an axiom that restricts the definition of sets by properties. The scope of a property P (A), like P (A) A/ A, must itself be a set, say B. Thus, instead of R = A A/ A we define≡ for any∈ set B the set R(B) = A A B, A / A . -
SET THEORY 1. the Suslin Problem 1.1. the Suslin Hypothesis. Recall
SET THEORY MOSHE KAMENSKY Abstract. Notes on set theory, mainly forcing. The first four sections closely follow the lecture notes Williams [8] and the book Kunen [4]. The last section covers topics from various sources, as indicated there. Hopefully, all errors are mine. 1. The Suslin problem 1.1. The Suslin hypothesis. Recall that R is the unique dense, complete and separable order without endpoints (up to isomorphism). It follows from the sepa- rability that any collection of pairwise disjoint open intervals is countable. Definition 1.1.1. A linear order satisfies the countable chain condition (ccc) if any collection of pairwise disjoint open intervals is countable. Hypothesis 1.1.2 (The Suslin hypothesis). (R; <) is the unique complete dense linear order without endpoints that satisfies the countable chain condition. We will not prove the Suslin hypothesis (this is a theorem). Instead, we will reformulate it in various ways. First, we have the following apparent generalisation of the Suslin hypothesis. Theorem 1.1.3. The following are equivalent: (1) The Suslin hypothesis (2) Any ccc linear order is separable Proof. Let (X; <) be a ccc linear order that is not separable. Assume first that it is dense. Then we may assume it has no end points (by dropping them). The completion again has the ccc, so by the Suslin hypothesis is isomorphic to R. Hence we may assume that X ⊆ R, but then X is separable. It remains to produce a dense counterexample from a general one. Define x ∼ y if both (x; y) and (y; x) are separable. This is clearly a convex equivalence relation, so the quotient Y is again a linear order satisfying the ccc. -
Initial Embeddings in the Surreal Number Tree A
INITIAL EMBEDDINGS IN THE SURREAL NUMBER TREE A Thesis Presented to The Honors Tutorial College Ohio University In Partial Fulfillment of the Requirements for Graduation from the Honors Tutorial College with the degree of Bachelor of Science in Mathematics by Elliot Kaplan April 2015 Contents 1 Introduction 1 2 Preliminaries 3 3 The Structure of the Surreal Numbers 11 4 Leaders and Conway Names 24 5 Initial Groups 33 6 Initial Integral Domains 43 7 Open Problems and Closing Remarks 47 1 Introduction The infinitely large and the infinitely small have both been important topics throughout the history of mathematics. In the creation of modern calculus, both Isaac Newton and Gottfried Leibniz relied heavily on the use of infinitesimals. However, in 1887, Georg Cantor attempted to prove that the idea of infinitesimal numbers is a self-contradictory concept. This proof was valid, but its scope was sig- nificantly narrower than Cantor had thought, as the involved assumptions limited the proof's conclusion to only one conception of number. Unfortunately, this false claim that infinitesimal numbers are self-contradictory was disseminated through Bertrand Russell's The Principles of Mathematics and, consequently, was widely believed throughout the first half of the 20th century [6].Of course, the idea of in- finitesimals was never entirely abandoned, but they were effectively banished from calculus until Abraham Robinson's 1960s development of non-standard analysis, a system of mathematics which incorporated infinite and infinitesimal numbers in a mathematically rigorous fashion. One non-Archimedean number system is the system of surreal numbers, a sys- tem first constructed by John Conway in the 1970s [2]. -
Searching in Dynamic Tree-Like Partial Orders
Searching in Dynamic Tree-Like Partial Orders Brent Heeringa1, Marius C˘at˘alinIordan2, and Louis Theran3 1 Dept. of Computer Science. Williams College. [email protected]? 2 Dept. of Computer Science. Stanford University. [email protected]?? 3 Dept. of Mathematics. Temple University. [email protected]??? Abstract. We give the first data structure for the problem of main- taining a dynamic set of n elements drawn from a partially ordered uni- verse described by a tree. We define the Line-Leaf Tree, a linear-sized data structure that supports the operations: insert; delete; test member- ship; and predecessor. The performance of our data structure is within an O(log w)-factor of optimal. Here w ≤ n is the width of the partial- order|a natural obstacle in searching a partial order. 1 Introduction A fundamental problem in data structures is maintaining an ordered set S of n items drawn from a universe U of size M n. For a totally ordered U, the dictionary operations: insert; delete; test membership; and predecessor are all supported in O(log n) time and O(n) space in the comparison model via balanced binary search trees. Here we consider the relaxed problem where U is partially ordered and give the first data structure for maintaining a dynamic partially ordered set drawn from a universe that can be described by a tree. As a motivating example, consider an email user that has stockpiled years of messages into a series of hierarchical folders. When searching for an old message, filing away a new message, or removing an impertinent message, the user must navigate the hierarchy. -
Some Results in the Extension with a Coherent Suslin Tree
Some results in the extension with a coherent Suslin tree Teruyuki Yorioka (Shizuoka University) Winter School in Abstract Analysis section Set Theory & Topology 28th January { 4th February, 2012 Motivation Theorem (Kunen, Rowbottom, Solovay, etc). implies K : Every ccc forcing MA@1 2 has property K. Question (Todorˇcevi´c). Does K imply ? 2 MA@1 Theorem (Todorˇcevi´c). PID +p > @1 implies no S-spaces. Question (Todorˇcevi´c). Under PID, does no S-spaces imply p > @1? Definition (Todorˇcevi´c). PFA(S) is an axiom that there exists a coherent Suslin tree S such that the forcing axiom holds for every proper forcing which preserves S to be Suslin. Theorem (Farah). t = @1 holds in the extension with a Suslin tree. Proof. Suppose that T is a Suslin tree, and take π : T ! [!]@0 such that ∗ s ≤T t ! π(s) ⊇ π(t) and s ?T t ! π(s) \ π(t) finite: Then for a generic branch G through T , the set fπ(s): s 2 Gg is a ⊆∗-decreasing sequence which doesn't have its lower bound in [!]@0. Motivation Theorem (Kunen, Rowbottom, Solovay, etc). implies K : Every ccc forcing MA@1 2 has property K. Question (Todorˇcevi´c). Does K imply ? 2 MA@1 Theorem (Todorˇcevi´c). PID +p > @1 implies no S-spaces. Question (Todorˇcevi´c). Under PID, does no S-spaces imply p > @1? Definition (Todorˇcevi´c). PFA(S) is an axiom that there exists a coherent Suslin tree S such that the forcing axiom holds for every proper forcing which preserves S to be Suslin. Question (Todorˇcevi´c).