scholarly journals Embeddability Between Orderings and GCH

2021 ◽  
Vol 56 ◽  
pp. 101-109
Author(s):  
Rodrigo A. Freire
Keyword(s):  
Structure Theorem ◽  
Linear Ordering ◽  
Linear Orderings

We provide some statements equivalent in ZFC to GCH, and also to GCH above a given cardinal. These statements express the validity of the notions of replete and well-replete car- dinals, which are introduced and proved to be specially relevant to the study of cardinal exponentiation. As a byproduct, a structure theorem for linear orderings is proved to be equivalent to GCH: for every linear ordering L, at least one of L and its converse is universal for the smaller well-orderings.

A Model of the Real Numbers

1963 ◽  
Vol 6 (2) ◽  
pp. 239-255
Author(s):  
Stanton M. Trott
Keyword(s):  
Irrational Number ◽  
Additive Group ◽  
Linear Ordering ◽  
Real Numbers ◽  
Ordered Group ◽  
The Real ◽  
Linear Orderings ◽  
Positive Elements ◽  
The Law ◽  
Ordered Pairs

The model of the real numbers described below was suggested by the fact that each irrational number ρ determines a linear ordering of J2, the additive group of ordered pairs of integers. To obtain the ordering, we define (m, n) ≤ (m', n') to mean that (m'- m)ρ ≤ n' - n. This order is invariant with group translations, and hence is called a "group linear ordering". It is completely determined by the set of its "positive" elements, in this case, by the set of integer pairs (m, n) such that (0, 0) ≤ (m, n), or, equivalently, mρ < n. The law of trichotomy for linear orderings dictates that only the zero of an ordered group can be both positive and negative.

On Ehrenfeucht-Fraïssé equivalence of linear orderings

1990 ◽  
Vol 55 (1) ◽  
pp. 65-73 ◽  
Author(s):  
Juha Oikkonen
Keyword(s):  
Binary Relation ◽  
Linear Ordering ◽  
Relation Symbol ◽  
Linear Orderings ◽  
Binary Relation Symbol

AbstractC. Karp has shown that if α is an ordinal with ωα = α and A is a linear ordering with a smallest element, then α and α ⊗ A are equivalent in L∞ω up to quantifer rank α. This result can be expressed in terms of Ehrenfeucht-Fraïssé games where player ∀ has to make additional moves by choosing elements of a descending sequence in α. Our aim in this paper is to prove a similar result for Ehrenfeucht-Fraïssé games of length ω1. One implication of such a result will be that a certain infinite quantifier language cannot say that a linear ordering has no descending ω1-sequences (when the alphabet contains only one binary relation symbol). Connected work is done by Hyttinen and Oikkonen in [H] and [O].

Uniformly defined descending sequences of degrees

1976 ◽  
Vol 41 (2) ◽  
pp. 363-367 ◽  
Author(s):  
Harvey Friedman
Keyword(s):  
Unique Solution ◽  
Set Theory ◽  
Limit Point ◽  
Infinite Sequence ◽  
Linear Ordering ◽  
Natural Numbers ◽  
Linear Orderings ◽  
Nonstandard Models ◽  
Related Paper ◽  
Models Of Set Theory

This paper answers some questions which naturally arise from the Spector-Gandy proof of their theorem that the π11 sets of natural numbers are precisely those which are defined by a Σ11 formula over the hyperarithmetic sets. Their proof used hierarchies on recursive linear orderings (H-sets) which are not well orderings. (In this respect they anticipated the study of nonstandard models of set theory.) The proof hinged on the following fact. Let e be a recursive linear ordering. Then e is a well ordering if and only if there is an H-set on e which is hyperarithmetic. It was implicit in their proof that there are recursive linear orderings which are not well orderings, on which there are H-sets. Further information on such nonstandard H-sets (often called pseudohierarchies) can be found in Harrison [4]. It is natural to ask: on which recursive linear orderings are there H-sets?In Friedman [1] it is shown that there exists a recursive linear ordering e that has no hyperarithmetic descending sequences such that no H-set can be placed on e. In [1] it is also shown that if e is a recursive linear ordering, every point of which has an immediate successor and either has finitely many predecessors or is finitely above a limit point (heretofore called adequate) such that an H-set can be placed on e, then e has no hyperarithmetic descending sequences. In a related paper, Friedman [2] shows that there is no infinite sequence xn of codes for ω-models of the arithmetic comprehension axiom scheme such that each xn+ 1 is a set in the ω-model coded by xn, and each xn+1 is the unique solution of P(xn, xn+1) for some fixed arithmetic P.

On the Equimorphism Types of Linear Orderings

2007 ◽  
Vol 13 (1) ◽  
pp. 71-99 ◽  
Author(s):  
Antonio Montalbán
Keyword(s):  
Equivalence Relation ◽  
Reverse Mathematics ◽  
Computability Theory ◽  
Equivalence Classes ◽  
Linear Ordering ◽  
Partially Ordered ◽  
Linear Orderings ◽  
Definition Of

§1. Introduction. A linear ordering (also known astotal ordering) embedsinto another linear ordering if it is isomorphic to a subset of it. Two linear orderings are said to beequimorphicif they can be embedded in each other. This is an equivalence relation, and we call the equivalence classesequimorphism types. We analyze the structure of equimorphism types of linear orderings, which is partially ordered by the embeddability relation. Our analysis is mainly fromthe viewpoints of Computability Theory and Reverse Mathematics. But we also obtain results, as the definition of equimorphism invariants for linear orderings, which provide a better understanding of the shape of this structure in general.This study of linear orderings started by analyzing the proof-theoretic strength of a theorem due to Jullien [Jul69]. As is often the case in Reverse Mathematics, to solve this problem it was necessary to develop a deeper understanding of the objects involved. This led to a variety of results on the structure of linear orderings and the embeddability relation on them. These results can be divided into three groups.

scholarly journals ALGEBRAIC LINEAR ORDERINGS

2011 ◽  
Vol 22 (02) ◽  
pp. 491-515 ◽  
Author(s):  
S. L. BLOOM ◽  
Z. ÉSIK
Keyword(s):  
Order Type ◽  
Linear Ordering ◽  
Type Result ◽  
First Order ◽  
Linear Orderings ◽  
Context Free Language ◽  
Categorical Algebra ◽  
Context Free ◽  
Free Language

An algebraic linear ordering is a component of the initial solution of a first-order recursion scheme over the continuous categorical algebra of countable linear orderings equipped with the sum operation and the constant 1. Due to a general Mezei-Wright type result, algebraic linear orderings are exactly those isomorphic to the linear ordering of the leaves of an algebraic tree. Using Courcelle's characterization of algebraic trees, we obtain the fact that a linear ordering is algebraic if and only if it can be represented as the lexicographic ordering of a deterministic context-free language. When the algebraic linear ordering is a well-ordering, its order type is an algebraic ordinal. We prove that the Hausdorff rank of any scattered algebraic linear ordering is less than ωω. It follows that the algebraic ordinals are exactly those less than ωωω.

scholarly journals On computable self-embeddings of computable linear orderings

2009 ◽  
Vol 74 (4) ◽  
pp. 1352-1366 ◽  
Author(s):  
Rodney G. Downey ◽  
Bart Kastermans ◽  
Steffen Lempp
Keyword(s):  
Open Problem ◽  
Linear Ordering ◽  
Linear Orderings ◽  
Precise Characterization

AbstractWe solve a longstanding question of Rosenstein, and make progress toward solving a long-standing open problem in the area of computable linear orderings by showing that every computable η-like linear ordering without an infinite strongly η-like interval has a computable copy without nontrivial computable self-embedding.The precise characterization of those computable linear orderings which have computable copies without nontrivial computable self-embedding remains open.

Two further combinatorial theorems equivalent to the 1-consistency of peano arithmetic

1983 ◽  
Vol 48 (4) ◽  
pp. 1090-1104 ◽  
Author(s):  
Peter Clote ◽  
Kenneth Mcaloon
Keyword(s):  
Peano Arithmetic ◽  
Infinite Subset ◽  
Linear Ordering ◽  
Tree Property ◽  
Preceding Theorem ◽  
Linear Orderings ◽  
Finite Set ◽  
The One ◽  
König’S Lemma

We give two new finite combinatorial statements which are independent of Peano arithmetic, using the methods of Kirby and Paris [6] and Paris [12]. Both are in fact equivalent over Peano arithmetic (denoted by P) to its 1-consistency. The first involves trees and the second linear orderings. Both were “motivated” by anti-basis theorems of Clote (cf. [1], [2]). The one involving trees, however, is not unrelated to the Kirby-Paris characterization of strong cuts in terms of the tree property [6], but, in fact, comes directly from König's lemma, of which it is a miniaturization. (See the remark preceding Theorem 3 below.) The resulting combinatorial statement is easily seen to imply the independent statement discovered by Mills [11], but it is not clear how to show their equivalence over Peano arithmetic without going through 1-consistency. The one involving linear orderings miniaturizes the property of infinite sets X that any linear ordering of X is isomorphic to ω or ω* on some infinite subset of X. Both statements are analogous to Example 2 of [12] and involve the notion of dense [12] or relatively large [14] finite set.We adopt the notations and definitions of [6] and [12]. We shall in particular have need of the notions of semiregular, regular and strong initial segments and of indicators.

Elementary embedding between countable Boolean algebras

1991 ◽  
Vol 56 (4) ◽  
pp. 1212-1229
Author(s):  
Robert Bonnet ◽  
Matatyahu Rubin
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Complete Theory ◽  
Linear Ordering ◽  
Elementary Embedding ◽  
Linear Orderings ◽  
Partial Orderings ◽  
Atomless Boolean Algebra ◽  
Atomic Boolean Algebra ◽  
Quasi Ordering

AbstractFor a complete theory of Boolean algebras T, let MT denote the class of countable models of T. For B1, B2 ∈ MT, let B1 ≤ B2 mean that B2 is elementarily embeddable in B2. Theorem 1. For every complete theory of Boolean algebras T, if T ≠ Tω, then ‹MT, ≤› is well-quasi-ordered. ∎ We define Tω. For a Boolean algebra B, let I(B) be the ideal of all elements of the form a + s such that B ↾ a is an atomic Boolean algebra and B ↾ s is an atomless Boolean algebra. The Tarski derivative of B is defined as follows: B(0) = B and B(n + 1) = B(n)/I(B(n)). Define Tω to be the theory of all Boolean algebras such that for every n ∈ ω, B(n) ≠ {0}. By Tarski [1949], Tω is complete. Recall that ‹A, < › is partial well-quasi-ordering, it is a partial quasi-ordering and for every {ai, ⃒ i ∈ ω} ⊆ A, there are i < j < ω such that ai ≤ aj. Theorem 2. contains a subset M such that the partial orderings ‹M, ≤ ↾ M› and are isomorphic. ∎ Let M′0 denote the class of all countable Boolean algebras. For B1, B2 ∈ M′0, let B1 ≤′ B2 mean that B1 is embeddable in B2. Remark. ‹M′0, ≤′› is well-quasi-ordered. ∎ This follows from Laver's theorem [1971] that the class of countable linear orderings with the embeddability relation is well-quasi-ordered and the fact that every countable Boolean algebra is isomorphic to a Boolean algebra of a linear ordering.

Two-dimensional partial orderings: Recursive model theory

1980 ◽  
Vol 45 (1) ◽  
pp. 121-132 ◽  
Author(s):  
Alfred B. Manaster ◽  
Joseph G. Rosenstein
Keyword(s):  
Model Theory ◽  
Companion Paper ◽  
Partial Ordering ◽  
Linear Ordering ◽  
Two Dimensional ◽  
Recursive Model ◽  
Linear Orderings ◽  
Rational Plane ◽  
Partial Orderings

In this paper and the companion paper [9] we describe a number of contrasts between the theory of linear orderings and the theory of two-dimensional partial orderings.The notion of dimensionality for partial orderings was introduced by Dushnik and Miller [3], who defined a partial ordering 〈A, R〉 to be n-dimensional if there are n linear orderings of A, 〈A, L1〉, 〈A, L2〉 …, 〈A, Ln〉 such that R = L1 ∩ L2 ∩ … ∩ Ln. Thus, for example, if Q is the linear ordering of the rationals, then the (rational) plane Q × Q with the product ordering (〈x1, y1〉 ≤Q×Q 〈x2, y2, if and only if x1 ≤ x2 and y1 ≤ y2) is 2-dimensional, since ≤Q×Q is the intersection of the two lexicographic orderings of Q × Q. In fact, as shown by Dushnik and Miller, a countable partial ordering is n-dimensional if and only if it can be embedded as a subordering of Qn.Two-dimensional partial orderings have attracted the attention of a number of combinatorialists in recent years. A basis result recently obtained, independently, by Kelly [7] and Trotter and Moore [10], describes explicitly a collection of finite partial orderings such that a partial ordering is a 2dpo if and only if it contains no element of as a subordering.

scholarly journals ON THE COMPLEXITY OF THE SUCCESSIVITY RELATION IN COMPUTABLE LINEAR ORDERINGS

2010 ◽  
Vol 10 (01n02) ◽  
pp. 83-99 ◽  
Author(s):  
ROD DOWNEY ◽  
STEFFEN LEMPP ◽  
GUOHUA WU
Keyword(s):  
New Method ◽  
Independent Interest ◽  
Linear Ordering ◽  
Turing Degrees ◽  
Linear Orderings ◽  
Open Question ◽  
Computably Enumerable

In this paper, we solve a long-standing open question (see, e.g. Downey [6, Sec. 7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering [Formula: see text] has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees. To do this, we use a new method of constructing [Formula: see text]-isomorphisms, which has already found other applications such as Downey, Kastermans and Lempp [9] and is of independent interest. It would seem to promise many further applications.

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