On properties of theories which preclude the existence of universal models

We introduce the oak property of first order theories, which is a syntactical condition that we show to be sufficient for a theory not to have universal models in cardinality ? when certain cardinal arithmetic assumptions about ? implying the failure of GCH (and close to the failure of SCH) hold. We give two examples of theories that have the oak property and show that none of these examples satisfy SOP4, not even SOP3. This is related to the question of the connection of the property SOP4 to non-universality, as was raised by the earlier work of Shelah. One of our examples is the theory View the MathML source for which non-universality results similar to the ones we obtain are already known; hence we may view our results as an abstraction of the known results from a concrete theory to a class of theories. We show that no theory with the oak property is simple

Chain models, trees of singular cardinality and dynamic EF games

Let κ be a singular cardinal. Karp's notion of a chain model of size ? is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf(κ). With a notion of satisfaction and (chain)-isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game is associated a tree which is a tree of size κ with no κ-branches (even no cf(κ)-branches). The measure of how non-isomorphic the models are is reflected by a certain order on these trees, called reduction. We study the collection of trees of size κ with no κ-branches under this notion and prove that when cf(κ) = ω this collection is rather regular; in particular it has universality number exactly κ+. Such trees are then used to develop a descriptive set theory of the space cf(κ)κ.The main result of the paper gives in the case of κ strong limit singular an exact connection between the descriptive set-theoretic complexity of the chain isomorphism orbit of a model, the reduction order on the trees and winning strategies in the corresponding dynamic EF games. In particular we obtain a neat analog of the notion of Scott watershed from the Scott analysis of countable models

Extension and reconstruction theorems for the Urysohn universal metric space

We prove some extension theorems involving uniformly continuous maps of the universal Urysohn space. We also prove reconstruction theorems for certain groups of autohomeomorphisms of this space and of its open subsets.Comment: Final and shortened version, 25 pages, to appear in Czechoslovak Math.

Tree indiscernibilities, revisited

We give definitions that distinguish between two notions of indiscernibility for a set \{a_\eta \mid \eta \in \W\} that saw original use in \cite{sh90}, which we name \textit{\s-} and \textit{\n-indiscernibility}. Using these definitions and detailed proofs, we prove \s- and \n-modeling theorems and give applications of these theorems. In particular, we verify a step in the argument that TP is equivalent to TP$_1$ or TP$_2$ that has not seen explication in the literature. In the Appendix, we exposit the proofs of \citep[{App. 2.6, 2.7}]{sh90}, expanding on the details.Comment: submitte

Club does not imply the existence of a Suslin tree

We prove that club does not imply the existence of a Suslin tree, so answering a question of I. Juhasz

The experience and perceptions of gender discriminination within the family: Impact on basic self-esteem and coping strategies

Iako se žene u Srbiji i dalje suočavaju s različitim oblicima diskriminacije u porodičnim i partnerskim odnosima, one sebe retko opažaju kao žrtve takvog postupanja. Umanjivanje opažene rodne diskriminacije pomaže da kratkoročno očuvaju pozitivnu sliku o sebi, ali ih istovremeno sprečava da reaguju na činove diskriminacije i dugoročno zaštite svoju psihološku dobrobit. Stoga u ovom radu ispitujemo diskrepanciju između ličnih iskustava i subjektivnog opažanja rodne diskriminacije u porodičnim i partnerskim odnosima, kao i povezanost opažanja ovakvog tretmana sa bazičnim samopoštovanjem i strategijama suočavanja sa postupcima rodne diskriminacije. Podaci su prikupljeni na prigodnom uzorku od 273 žene starosti od 19 do 62 godine, iz različitih krajeva Srbije, pomoću sledećih instrumenata: percepcija lične rodne diskriminacije, Rozenbergova skala samopoštovanja, Lista rodno diskriminatornih događaja i Suočavanje sa rodnom diskriminacijom. Nalazi potvrđuju diskrepanciju između nivoa opažene i doživljene rodne diskriminacije. Naime, iako žene smatraju da nisu bile diskriminisane, njihova iskustva svedoče o učestaloj diskriminaciji u porodičnim i partnerskim odnosima. Rezultati ukazuju da je opažanje lične diskriminacije u porodici porekla povezano sa nižim samopoštovanjem žena, kao i sa korišćenjem pasivnih strategija suočavanja sa diskriminacijom u odraslom dobu. Istovremeno, očuvano samopoštovanje predstavlja jedinu varijablu merenu u ovoj studiji koja predviđa upotrebu aktivnih strategija suočavanja sa ovakvim iskustvima. Nalazi potvrđuju psihološku dinamiku začaranog kruga diskriminacije u porodičnim odnosima: prepoznavanje rodne diskriminacije narušava samopoštovanje žene i pasivizuje je, čineći je podložnom novim i težim činovima diskriminacije koji dalje produbljuju nisko samopštovanje i pasivnost. Zbog toga su za prevenciju rodno zasnovanog nasilja neophodni edukativni programi u cilju senzibilisanja javnosti za prepoznavanje različitih i diskretnijih ispoljavanja rodne diskriminacije u svim sferama života: Although women in Serbia still face different forms of discrimination in family and intimate partner relationships, they rarely perceive themselves as victims of such discrimination. Minimising the observed gender discrimination helps them to maintain a positive self image in the short run; however, in the long haul it prevents them from reacting to such acts and protecting their psychological wellbeing. In this paper we examine the discrepancy between personal experiences and subjective perception of gender discrimination in family and intimate partner relationships, as well as the connection between the perception of such treatment and basic self-esteem and strategies of coping with gender discrimination. Data are collected from a suitable sample of 273 women aged between 19 and 62, from different parts of Serbia, using the following instruments: The perception of personal gender discrimination, Rosenberg’s self-esteem scale, The list of instances of gender discrimination, and Coping with gender discrimination. The findings confirm the discrepancy between the level of perceived and experienced gender discrimination. Although women do not see themselves as discriminated, their experiences testify to frequent discrimination in family and intimate partner relationships. The results indicate the correlation between the perception of personal discrimination within the family and the women’s lower self-esteem in adulthood. At the same time, the preserved self-esteem is one of the variables measured in this study, which presupposes the use of active strategies in coping with such experiences. The findings confirm the psychological dynamics of vicious cycle of discrimination within family relationships: recognising gender discrimination undermines a woman’s self-esteem and passivises her; it makes her liable to new and more severe acts of discrimination which then further undermines her self-esteem and contributes to her passivity. Educational programmes are essential in the prevention of gender based violence, aiming to raise awareness in the general public of the diverse and more discrete forms of gender discrimination in all spheres of life

A new foundational crisis in mathematics, is it really happening?

The article reconsiders the position of the foundations of mathematics after the discovery of HoTT. Discussion that this discovery has generated in the community of mathematicians, philosophers and computer scientists might indicate a new crisis in the foundation of mathematics. By examining the mathematical facts behind HoTT and their relation with the existing foundations, we conclude that the present crisis is not one. We reiterate a pluralist vision of the foundations of mathematics. The article contains a short survey of the mathematical and historical background needed to understand the main tenets of the foundational issues.Comment: Final versio

Universality of uniform Eberlein compacta

We prove that if $\mu^+ 2^{\aleph_0}$, then there is no family of less than $\mu^{\aleph_0}$ c-algebras of size $\lambda$ which are jointly universal for c-algebras of size $\lambda$. On the other hand, it is consistent to have a cardinal $\lambda\ge \aleph_1$ as large as desired and satisfying \lambda^{\lambda^{++}, while there are $\lambda^{++}$ c-algebras of size $\lambda^+$ that are jointly universal for c-algebras of size $\lambda^+$. Consequently, by the known results of M. Bell, it is consistent that there is $\lambda$ as in the last statement and $\lambda^{++}$ uniform Eberlein compacta of weight $\lambda^+$ such that at least one among them maps onto any Eberlein compact of weight $\lambda^+$ (we call such a family universal). The only positive universality results for Eberlein compacta known previously required the relevant instance of $GCH$ to hold. These results complete the answer to a question of Y. Benyamini, M. E. Rudin and M. Wage from 1977 who asked if there always was a universal uniform Eberlein compact of a given weight