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

On Formalism Freeness : Implementing Gödel's 1946 Princeton Bicentennial Lecture

Peer reviewe

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

Contributions to the theory of Large Cardinals through the method of Forcing

[eng] The present dissertation is a contribution to the field of Mathematical Logic and, more particularly, to the subfield of Set Theory. Within Set theory, we are mainly concerned with the interactions between the largecardinal axioms and the method of Forcing. This is the line of research with a deeper impact in the subsequent configuration of modern Mathematics. This area has found many central applications in Topology [ST71][Tod89], Algebra [She74][MS94][DG85][Dug85], Analysis [Sol70] or Category Theory [AR94][Bag+15], among others. The dissertation is divided in two thematic blocks: In Block I we analyze the large-cardinal hierarchy between the first supercompact cardinal and Vopenka’s Principle (Part I). In Block II we make a contribution to Singular Cardinal Combinatorics (Part II and Part III). Specifically, in Part I we investigate the Identity Crisis phenomenon in the region comprised between the first supercompact cardinal and Vopenka’s Principle. As a result, we settle all the questions that were left open in [Bag12, §5]. Afterwards, we present a general theory of preservation of C(n)– extendible cardinals under class forcing iterations from which we derive many applications. In Part II and Part III we analyse the relationship between the Singular Cardinal Hypothesis (SCH) and other combinatorial principles, such as the tree property or the reflection of stationary sets. In Part II we generalize the main theorems of [FHS18] and [Sin16] and manage to weaken the largecardinal hypotheses necessary for Magidor-Shelah’s theorem [MS96]. Finally, in Part III we introduce the concept of _-Prikry forcing as a generalization of the classical notion of Prikry-type forcing. Subsequently we devise an abstract iteration scheme for this family of posets and, as an application, we prove the consistency of ZFC + ¬SCH_ + Refl([cat] La present tesi és una contribució a l’estudi de la Lògica Matemàtica i més particularment a la Teoria de Conjunts. Dins de la Teoria de Conjunts, la nostra àrea de recerca s’emmarca dins l’estudi de les interaccions entre els Axiomes de Grans Cardinals i el mètode de Forcing. Aquestes dues eines han tigut un impacte molt profund en la configuració de la matemàtica contemporànea com a conseqüència de la resolució de qüestions centrals en Topologia [ST71][Tod89], Àlgebra [She74][MS94][DG85][Dug85], Anàlisi Matemàtica [Sol70] o Teoria de Categories [AR94][Bag+15], entre d’altres. La tesi s’articula entorn a dos blocs temàtics. Al Bloc I analitzem la jerarquia de Grans Cardinals compresa entre el primer cardinal supercompacte i el Principi de Vopenka (Part I), mentre que al Bloc II estudiem alguns problemes de la Combinatòria Cardinal Singular (Part II i Part III). Més precisament, a la Part I investiguem el fenòmen de Crisi d’Identitat en la regió compresa entre el primer cardinal supercompacte i el Principi de Vopenka. Com a conseqüència d’aquesta anàlisi resolem totes les preguntes obertes de [Bag12, §5]. Posteriorment presentem una teoria general de preservació de cardinals C(n)–extensibles sota iteracions de longitud ORD, de la qual en derivem nombroses aplicacions. A la Part II i Part III analitzem la relació entre la Hipòtesi dels Cardinals Singulars (SCH) i altres principis combinatoris, tals com la Propietat de l’Arbre o la reflexió de conjunts estacionaris. A la Part II obtenim sengles generalitzacions dels teoremes principals de [FHS18] i [Sin16] i afeblim les hipòtesis necessàries perquè el teorema de Magidor-Shelah [MS96] siga cert. Finalment, a la Part III, introduïm el concepte de forcing _-Prikry com a generalització de la noció clàssica de forcing del tipus Prikry. Posteriorment dissenyem un esquema d’iteracions abstracte per aquesta família de forcings i, com a aplicació, derivem la consistència de ZFC + ¬SCH_ + Refl

The Alor-Pantar languages: History and typology (Second edition)

The Alor-Pantar family constitutes the westernmost outlier group of Papuan (Non-Austronesian) languages. Its twenty or so languages are spoken on the islands of Alor and Pantar, located just north of Timor, in eastern Indonesia. Together with the Papuan languages of Timor, they make up the Timor-Alor-Pantar family. The languages average 5,000 speakers and are under pressure from the local Malay variety as well as the national language, Indonesian. This volume studies the internal and external linguistic history of this interesting group, and showcases some of its unique typological features, such as the preference to index the transitive patient-like argument on the verb but not the agent-like one; the extreme variety in morphological alignment patterns; the use of plural number words; the existence of quinary numeral systems; the elaborate spatial deictic systems involving an elevation component; and the great variation exhibited in their kinship systems. Unlike many other Papuan languages, Alor-Pantar languages do not exhibit clause-chaining, do not have switch reference systems, never suffix subject indexes to verbs, do not mark gender, but do encode clusivity in their pronominal systems. Indeed, apart from a broadly similar head-final syntactic profile, there is little else that the Alor-Pantar languages share with Papuan languages spoken in other regions. While all of them show some traces of contact with Austronesian languages, in general, borrowing from Austronesian has not been intense, and contact with Malay and Indonesian is a relatively recent phenomenon in most of the Alor-Pantar region. This is the second edition of the volume that was originally published in 2014. In this edition, typographical errors have been corrected, small textual improvements have been implemented, broken URL links repaired or removed, and references updated. The overall content of the chapters has not been changed