Notes on cardinals that are characterizable by a complete (Scott) sentence

This is part I of a study on cardinals that are characterizable by Scott sentences. Building on [3], [6] and [1] we study which cardinals are characterizable by a Scott sentence $\phi$, in the sense that $\phi$ characterizes $\kappa$, if $\phi$ has a model of size $\kappa$, but no models of size $\kappa^+$. We show that the set of cardinals that are characterized by a Scott sentence is closed under successors, countable unions and countable products (cf. theorems 2.3, 3.4, and corollary 3.6). We also prove that if $\aleph_\alpha$ is characterized by a Scott sentence, at least one of $\aleph_alpha$ and $\aleph_alpha^+$ is homogeneously characterizable (cf. definition 1.3 and theorem 2.9). Based on Shelah's [8], we give counterexamples that characterizable cardinals are not closed under predecessors, or cofinalities.Comment: Version 2 replaces version 1 of the same paper (with the same title), but version 2 contains only half of the content of version 1. The second half of version 1 will be posted by itsel

Linear Orderings and Powers of Characterizable Cardinal

The current paper answers an open question of abs/1007.2426 We say that a countable model M characterizes an infinite cardinal kappa, if the Scott sentence of M has a model in cardinality kappa, but no models in cardinality kappa plus. If M is linearly ordered by <, we will say that the linear ordering (M,<) characterizes kappa. It is known that if kappa is characterizable, then kappa plus is characterizable by a linear ordering. Also, if kappa is characterizable by a dense linear ordering with an increasing sequence of size kappa, then 2^kappa is characterizable. We show that if kappa is homogeneously characterizable, then kappa is characterizable by a dense linear ordering, while the converse fails. The main theorems are: 1) If kappa>2^lambda is a characterizable cardinal, lambda is characterizable by a dense linear ordering and lambda is the least cardinal such that kappa^lambda>kappa, then kappa^lambda is also characterizable, 2) if aleph_alpha and kappa^(aleph_alpha) are characterizable cardinals, then the same is true for kappa^(aleph_(alpha+beta)), for all countable beta. Combining these two theorems we get that if kappa>2^(aleph_alpha) is a characterizable cardinal, aleph_alpha is characterizable by a dense linear ordering and aleph_alpha is the least cardinal such that kappa^(aleph_alpha)>kappa, then for all beta<alpha+omega_1, kappa^(aleph_beta) is characterizable. Also if kappa is a characterizable cardinal, then kappa^(aleph_alpha) is characterizable, for all countable alpha.Comment: Version 01: 20 pages, no figures, submitted for publication in November 201

Characterizing the powerset by a complete (Scott) sentence

This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing the work from http://arxiv.org/abs/1007.2426v1. A cardinal $\kappa$ is characterized by a Scott sentence $\phi_M$, if $\phi_M$ has a model of size $\kappa$, but no model of $\kappa^+$. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if $\aleph_{\beta}$ is characterized by a Scott sentence, then $2^{\aleph_{\beta+\beta_1}}$ is (homogeneously) characterized by a Scott sentence, for all $0<\beta_1<\omega_1$. So, the answer to the above question is positive, except the case $\beta_1=0$ which remains open. As a consequence we derive that if $\alpha\le\beta$ and $\aleph_{\beta}$ is characterized by a Scott sentence, then $\aleph_{\alpha+\alpha_1}^{\aleph_{\beta+\beta_1}}$ is also characterized by a Scott sentence, for all $\alpha_1<\omega_1$ and $0<\beta_1<\omega_1$. Whence, depending on the model of ZFC, we see that the class of characterizable and homogeneously characterizable cardinals is much richer than previously known. Several open questions are also mentioned at the end.Comment: This paper is an updated version of the second half of version 1 of arXiv:1007.2426v

Topological Groups: Yesterday, Today, Tomorrow

In 1900, David Hilbert asked whether each locally euclidean topological group admits a Lie group structure. This was the fifth of his famous 23 questions which foreshadowed much of the mathematical creativity of the twentieth century. It required half a century of effort by several generations of eminent mathematicians until it was settled in the affirmative. These efforts resulted over time in the Peter-Weyl Theorem, the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups, and finally the solution of Hilbert 5 and the structure theory of locally compact groups, through the combined work of Andrew Gleason, Kenkichi Iwasawa, Deane Montgomery, and Leon Zippin. For a presentation of Hilbert 5 see the 2014 book “Hilbert’s Fifth Problem and Related Topics” by the winner of a 2006 Fields Medal and 2014 Breakthrough Prize in Mathematics, Terence Tao. It is not possible to describe briefly the richness of the topological group theory and the many directions taken since Hilbert 5. The 900 page reference book in 2013 “The Structure of Compact Groups” by Karl H. Hofmann and Sidney A. Morris, deals with one aspect of compact group theory. There are several books on profinite groups including those written by John S. Wilson (1998) and by Luis Ribes and ‎Pavel Zalesskii (2012). The 2007 book “The Lie Theory of Connected Pro-Lie Groups” by Karl Hofmann and Sidney A. Morris, demonstrates how powerful Lie Theory is in exposing the structure of infinite-dimensional Lie groups. The study of free topological groups initiated by A.A. Markov, M.I. Graev and S. Kakutani, has resulted in a wealth of interesting results, in particular those of A.V. Arkhangelʹskiĭ and many of his former students who developed this topic and its relations with topology. The book “Topological Groups and Related Structures” by Alexander Arkhangelʹskii and Mikhail Tkachenko has a diverse content including much material on free topological groups. Compactness conditions in topological groups, especially pseudocompactness as exemplified in the many papers of W.W. Comfort, has been another direction which has proved very fruitful to the present day

Formal approaches to number in Slavic and beyond (Volume 5)

The goal of this collective monograph is to explore the relationship between the cognitive notion of number and various grammatical devices expressing this concept in natural language with a special focus on Slavic. The book aims at investigating different morphosyntactic and semantic categories including plurality and number-marking, individuation and countability, cumulativity, distributivity and collectivity, numerals, numeral modifiers and classifiers, as well as other quantifiers. It gathers 19 contributions tackling the main themes from different theoretical and methodological perspectives in order to contribute to our understanding of cross-linguistic patterns both in Slavic and non-Slavic languages

On looking into words (and beyond): Structures, Relations, Analyses

On Looking into Words is a wide-ranging volume spanning current research into word structure and morphology, with a focus on historical linguistics and linguistic theory. The papers are offered as a tribute to Stephen R. Anderson, the Dorothy R. Diebold Professor of Linguistics at Yale, who is retiring at the end of the 2016-2017 academic year. The contributors are friends, colleagues, and former students of Professor Anderson, all important contributors to linguistics in their own right. As is typical for such volumes, the contributions span a variety of topics relating to the interests of the honorand. In this case, the central contributions that Anderson has made to so many areas of linguistics and cognitive science, drawing on synchronic and diachronic phenomena in diverse linguistic systems, are represented through the papers in the volume. The 26 papers that constitute this volume are unified by their discussion of the interplay between synchrony and diachrony, theory and empirical results, and the role of diachronic evidence in understanding the nature of language. Central concerns of the volume include morphological gaps, learnability, increases and declines in productivity, and the interaction of different components of the grammar. The papers deal with a range of linked synchronic and diachronic topics in phonology, morphology, and syntax (in particular, cliticization), and their implications for linguistic theory

On looking into words (and beyond): Structures, Relations, Analyses

On Looking into Words is a wide-ranging volume spanning current research into word structure and morphology, with a focus on historical linguistics and linguistic theory. The papers are offered as a tribute to Stephen R. Anderson, the Dorothy R. Diebold Professor of Linguistics at Yale, who is retiring at the end of the 2016-2017 academic year. The contributors are friends, colleagues, and former students of Professor Anderson, all important contributors to linguistics in their own right. As is typical for such volumes, the contributions span a variety of topics relating to the interests of the honorand. In this case, the central contributions that Anderson has made to so many areas of linguistics and cognitive science, drawing on synchronic and diachronic phenomena in diverse linguistic systems, are represented through the papers in the volume. The 26 papers that constitute this volume are unified by their discussion of the interplay between synchrony and diachrony, theory and empirical results, and the role of diachronic evidence in understanding the nature of language. Central concerns of the volume include morphological gaps, learnability, increases and declines in productivity, and the interaction of different components of the grammar. The papers deal with a range of linked synchronic and diachronic topics in phonology, morphology, and syntax (in particular, cliticization), and their implications for linguistic theory

On looking into words (and beyond): Structures, Relations, Analyses

On Looking into Words is a wide-ranging volume spanning current research into word structure and morphology, with a focus on historical linguistics and linguistic theory. The papers are offered as a tribute to Stephen R. Anderson, the Dorothy R. Diebold Professor of Linguistics at Yale, who is retiring at the end of the 2016-2017 academic year. The contributors are friends, colleagues, and former students of Professor Anderson, all important contributors to linguistics in their own right. As is typical for such volumes, the contributions span a variety of topics relating to the interests of the honorand. In this case, the central contributions that Anderson has made to so many areas of linguistics and cognitive science, drawing on synchronic and diachronic phenomena in diverse linguistic systems, are represented through the papers in the volume. The 26 papers that constitute this volume are unified by their discussion of the interplay between synchrony and diachrony, theory and empirical results, and the role of diachronic evidence in understanding the nature of language. Central concerns of the volume include morphological gaps, learnability, increases and declines in productivity, and the interaction of different components of the grammar. The papers deal with a range of linked synchronic and diachronic topics in phonology, morphology, and syntax (in particular, cliticization), and their implications for linguistic theory