scholarly journals On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1670
Author(s):  
Ali Enayat ◽  
Vladimir Kanovei ◽  
Vassily Lyubetsky
Keyword(s):  
Set Theory ◽  
Equivalence Relation ◽  
Lower Class ◽  
Equivalence Classes ◽  
The Other ◽  
Real Numbers ◽  
Generic Extensions ◽  
Models Of Set Theory

Examples of effectively indiscernible projective sets of real numbers in various models of set theory are presented. We prove that it is true, in Miller and Laver generic extensions of the constructible universe, that there exists a lightface Π21 equivalence relation on the set of all nonconstructible reals, having exactly two equivalence classes, neither one of which is ordinal definable, and therefore the classes are OD-indiscernible. A similar but somewhat weaker result is obtained for Silver extensions. The other main result is that for any n, starting with 2, the existence of a pair of countable disjoint OD-indiscernible sets, whose associated equivalence relation belongs to lightface Πn1, does not imply the existence of such a pair with the associated relation in Σn1 or in a lower class.

Cardinal collapsing and ordinal definability

1978 ◽  
Vol 43 (4) ◽  
pp. 635-642 ◽  
Author(s):  
Petr Štěpánek
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Axiom Of Choice ◽  
Complete Boolean Algebra ◽  
Ground Model ◽  
Inner Model ◽  
Definable Sets ◽  
The Axiom Of Choice ◽  
Generic Extensions ◽  
Models Of Set Theory

We shall describe Boolean extensions of models of set theory with the axiom of choice in which cardinals are collapsed by mappings definable from parameters in the ground model. In particular, starting from the constructible universe, we get Boolean extensions in which constructible cardinals are collapsed by ordinal definable sets.Let be a transitive model of set theory with the axiom of choice. Definability of sets in the generic extensions of is closely related to the automorphisms of the corresponding Boolean algebra. In particular, if G is an -generic ultrafilter on a rigid complete Boolean algebra C, then every set in [G] is definable from parameters in . Hence if B is a complete Boolean algebra containing a set of forcing conditions to collapse some cardinals in , it suffices to construct a rigid complete Boolean algebra C, in which B is completely embedded. If G is as above, then [G] satisfies “every set is -definable” and the inner model [G ∩ B] contains the collapsing mapping determined by B. To complete the result, it is necessary to give some conditions under which every cardinal from [G ∩ B] remains a cardinal in [G].The absolutness is granted for every cardinal at least as large as the saturation of C. To keep the upper cardinals absolute, it often suffices to construct C with the same saturation as B. It was shown in [6] that this is always possible, namely, that every Boolean algebra can be completely embedded in a rigid complete Boolean algebra with the same saturation.

An unpublished theorem of Solovay on OD partitions of reals into two non-OD parts, revisited

2020 ◽  
pp. 2150014 ◽  
Author(s):  
Ali Enayat ◽  
Vladimir Kanovei
Keyword(s):  
Equivalence Relation ◽  
Equivalence Classes ◽  
Generic Extensions

A definable pair of disjoint non-OD sets of reals (hence, indiscernible sets) exists in the Sacks and [Formula: see text]o-large generic extensions of the constructible universe L. More specifically, if [Formula: see text] is either Sacks generic or [Formula: see text]o generic real over L, then it is true in L[Formula: see text] that there is a lightface [Formula: see text] equivalence relation Q on the [Formula: see text] set [Formula: see text] with exactly two equivalence classes, and both those classes are non-OD sets.

scholarly journals Annihilator equivalence of torsion-free abelian groups

1992 ◽  
Vol 52 (1) ◽  
pp. 119-129
Author(s):  
P. Schultz ◽  
C. Vinsonhaler ◽  
W. J. Wickless
Keyword(s):  
Structural Properties ◽  
Equivalence Relation ◽  
Abelian Groups ◽  
Equivalence Classes ◽  
Pure Subgroup ◽  
The Other ◽  
Closure Properties ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractWe define an equivalence relation on the class of torsion-free abelian groups under which two groups are equivalent ifevery pure subgroup of one has a non-zero image in the other, and each has a non-zero image in every torsion-free factor of the other.We study the closure properties of the equivalence classes, and the structural properties of the class of all equivalence classes. Finally we identify a class of groups which satisfy Krull-Schmidt and Jordan-Hölder properties with respect to the equivalence.

Some applications of coarse inner model theory

1997 ◽  
Vol 62 (2) ◽  
pp. 337-365 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Equivalence Class ◽  
Set Theory ◽  
Model Theory ◽  
Equivalence Relation ◽  
Equivalence Classes ◽  
Descriptive Set Theory ◽  
Inner Model Theory ◽  
Inner Model ◽  
Descriptive Set

AbstractThe Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. determinacy implies that for every thin equivalence relation there is a real, N, over which every equivalence class is generic—and hence there is a good (N#) wellordering of the equivalence classes. Analogous results are obtained for and quasilinear orderings and determinacy is shown to imply that every prewellorder has rank less than .

Models of set theory with more real numbers than ordinals

1974 ◽  
Vol 39 (3) ◽  
pp. 579-583 ◽  
Author(s):  
Paul E. Cohen
Keyword(s):  
Standard Model ◽  
Set Theory ◽  
Axiom Of Choice ◽  
Infinite Cardinal ◽  
Real Numbers ◽  
Transitive Model ◽  
Lower Case ◽  
The Axiom Of Choice ◽  
Dependent Choice ◽  
Models Of Set Theory

Suppose M is a countable standard transitive model of set theory. P. J. Cohen [2] showed that if κ is an infinite cardinal of M then there is a one-to-one function Fκ from κ into the set of real numbers such that M[Fκ] is a model of set theory with the same cardinals as M.If Tκ is the range of Fκ then Cohen also showed [2] that M[Tκ] fails to satisfy the axiom of choice. We will give an easy proof of this fact.If κ, λ are infinite we will also show that M[Tκ] is elementarily equivalent to M[Tλ] and that (] in M[Fλ]) is elementarily equivalent to (] in M[FK]).Finally we show that there may be an N ∈ M[GK] which is a standard model of set theory (without the axiom of choice) and which has, from the viewpoint of M[GK], more real numbers than ordinals.We write ZFC and ZF for Zermelo-Fraenkel set theory, respectively with and without the axiom of choice (AC). GBC is Gödel-Bernays' set theory with AC. DC and ACℵo are respectively the axioms of dependent choice and of countable choice defined in [6].Lower case Greek characters (other than ω) are used as variables over ordinals. When α is an ordinal, R(α) is the set of all sets with rank less than α.

SINGULAR SWITCHED LINEAR SYSTEMS: SOME GEOMETRIC ASPECTS

2012 ◽  
Vol 26 (25) ◽  
pp. 1246006
Author(s):  
H. DIEZ-MACHÍO ◽  
J. CLOTET ◽  
M. I. GARCÍA-PLANAS ◽  
M. D. MAGRET ◽  
M. E. MONTORO
Keyword(s):  
Linear Systems ◽  
Group Action ◽  
Lie Group ◽  
Equivalence Relation ◽  
Geometric Approach ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Switched Linear Systems ◽  
Lie Group Action

We present a geometric approach to the study of singular switched linear systems, defining a Lie group action on the differentiable manifold consisting of the matrices defining their subsystems with orbits coinciding with equivalence classes under an equivalence relation which preserves reachability and derive miniversal (orthogonal) deformations of the system. We relate this with some new results on reachability of such systems.

ON THE PAIRWISE COMPATIBILITY PROPERTY OF SOME SUPERCLASSES OF THRESHOLD GRAPHS

2013 ◽  
Vol 05 (02) ◽  
pp. 1360002 ◽  
Author(s):  
TIZIANA CALAMONERI ◽  
ROSSELLA PETRESCHI ◽  
BLERINA SINAIMERI
Keyword(s):  
The Other ◽  
Real Numbers ◽  
Weighted Tree ◽  
Positive Edge ◽  
Threshold Graphs ◽  
Compatibility Graph ◽  
Unique Path ◽  
Compatibility Property

A graph G is called a pairwise compatibility graph (PCG) if there exists a positive edge weighted tree T and two non-negative real numbers d min and d max such that each leaf lu of T corresponds to a node u ∈ V and there is an edge (u, v) ∈ E if and only if d min ≤ dT (lu, lv) ≤ d max , where dT (lu, lv) is the sum of the weights of the edges on the unique path from lu to lv in T. In this paper we study the relations between the pairwise compatibility property and superclasses of threshold graphs, i.e., graphs where the neighborhoods of any couple of nodes either coincide or are included one into the other. Namely, we prove that some of these superclasses belong to the PCG class. Moreover, we tackle the problem of characterizing the class of graphs that are PCGs of a star, deducing that also these graphs are a generalization of threshold graphs.

STUDY OF THE INFLUENCE OF THE LEVEL OF ANXIETY DEVELOPMENT ON THE ACADEMIC PERFORMANCE OF PRIMARY SCHOOL STUDENTS

Vestnik ◽  
2021 ◽  
pp. 155-160
Author(s):  
А.Е. Малибаева ◽  
Б.К. Кайрат ◽  
А.И. Нуфтиева ◽  
Л.Б. Умбетьярова ◽  
М.С. Кулбаева ◽  
...  
Keyword(s):  
Academic Performance ◽  
Learning Outcomes ◽  
Learning Process ◽  
Child Anxiety ◽  
Lower Class ◽  
The Other ◽  
Primary School Students ◽  
School Students ◽  
The One ◽  
High Level

В современных стрессовых и негативных внешних экологических условиях растет число неуверенных в себе, эмоционально неустойчивых тревожных детей. В работах А.И.Захаровой, Н.В.Имеладзе, Л.М. Прихожановой говорится, что когда человек постоянно волнуется - возникает паника. Согласно анализу исследований многих авторов, детская тревога, с одной стороны, имеет психодинкамическую природу, с другой-является результатом социализации. По мнению психологов, у учащихся наблюдается высокий уровень тревожности в процессе обучения. В результате изучения данной проблемы установлено, что уровень тревожности и успеваемость ребенка тесно взаимосвязаны. Процесс приобщения детей, пришедших в школу, к процессу обучения тесно связан с процессом паники . In the current stressful and negative external environmental conditions, the number of insecure, emotionally unstable children with anxiety is growing. In the works of A.I. Zakharova, N.V. Imeladze, L.M. Prikhozhan, it is said that when a person is constantly agitated, panic occurs. According to the analysis of the research of many authors, child anxiety, on the one hand, has a psychodynamic nature, and on the other-is the result of socialization. According to psychologists, there is a high level of anxiety in students ' learning process. As a result of the study of this problem, it was found that the level of anxiety and the child's academic performance are closely related. The process of adaptation of children to the learning process is closely related to the panic process. However, the level of anxiety in lower-class students affects the learning process and learning outcomes.

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