{"title":"Local saturation and square everywhere","authors":"Monroe Eskew","doi":"10.1142/s0219061320500191","DOIUrl":"https://doi.org/10.1142/s0219061320500191","url":null,"abstract":"We show that it is consistent relative to a huge cardinal that for all infinite cardinals [Formula: see text], [Formula: see text] holds and there is a stationary [Formula: see text] such that [Formula: see text] is [Formula: see text]-saturated.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"7 12","pages":"2050019:1-2050019:33"},"PeriodicalIF":0.9,"publicationDate":"2019-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72370061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Approximate counting and NP search problems","authors":"L. Kolodziejczyk, Neil Thapen","doi":"10.1142/s021906132250012x","DOIUrl":"https://doi.org/10.1142/s021906132250012x","url":null,"abstract":"We study a new class of NP search problems, those which can be proved total using standard combinatorial reasoning based on approximate counting. Our model for this kind of reasoning is the bounded arithmetic theory [Formula: see text] of [E. Jeřábek, Approximate counting by hashing in bounded arithmetic, J. Symb. Log. 74(3) (2009) 829–860]. In particular, the Ramsey and weak pigeonhole search problems lie in the new class. We give a purely computational characterization of this class and show that, relative to an oracle, it does not contain the problem CPLS, a strengthening of PLS. As CPLS is provably total in the theory [Formula: see text], this shows that [Formula: see text] does not prove every [Formula: see text] sentence which is provable in bounded arithmetic. This answers the question posed in [S. Buss, L. A. Kołodziejczyk and N. Thapen, Fragments of approximate counting, J. Symb. Log. 79(2) (2014) 496–525] and represents some progress in the program of separating the levels of the bounded arithmetic hierarchy by low-complexity sentences. Our main technical tool is an extension of the “fixing lemma” from [P. Pudlák and N. Thapen, Random resolution refutations, Comput. Complexity, 28(2) (2019) 185–239], a form of switching lemma, which we use to show that a random partial oracle from a certain distribution will, with high probability, determine an entire computation of a [Formula: see text] oracle machine. The introduction to the paper is intended to make the statements and context of the results accessible to someone unfamiliar with NP search problems or with bounded arithmetic.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"23 1","pages":"2250012:1-2250012:31"},"PeriodicalIF":0.9,"publicationDate":"2018-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91368069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Locally definable subgroups of semialgebraic groups","authors":"E. Baro, Pantelis E. Eleftheriou, Y. Peterzil","doi":"10.1142/s0219061320500099","DOIUrl":"https://doi.org/10.1142/s0219061320500099","url":null,"abstract":"We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $Sigma_{i=1}^m(X-X)$ is an approximate subgroup of $G$.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"159 1","pages":"2050009:1-2050009:17"},"PeriodicalIF":0.9,"publicationDate":"2018-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76969515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"No Tukey reduction of Lebesgue null to Silver null sets","authors":"O. Spinas","doi":"10.1142/S0219061318500113","DOIUrl":"https://doi.org/10.1142/S0219061318500113","url":null,"abstract":"We prove that consistently the Lebesgue null ideal is not Tukey reducible to the Silver null ideal. This contrasts with the situation for the meager ideal which, by a recent result of the author, Spinas [Silver trees and Cohen reals, Israel J. Math. 211 (2016) 473–480] is Tukey reducible to the Silver ideal.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"23 1","pages":"1850011:1-1850011:32"},"PeriodicalIF":0.9,"publicationDate":"2018-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73756013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The canonical topology on dp-minimal fields","authors":"Will Johnson","doi":"10.1142/S0219061318500071","DOIUrl":"https://doi.org/10.1142/S0219061318500071","url":null,"abstract":"We construct a nontrivial definable type V field topology on any dp-minimal field [Formula: see text] that is not strongly minimal, and prove that definable subsets of [Formula: see text] have small boundary. Using this topology and its properties, we show that in any dp-minimal field [Formula: see text], dp-rank of definable sets varies definably in families, dp-rank of complete types is characterized in terms of algebraic closure, and [Formula: see text] is finite for all [Formula: see text]. Additionally, by combining the existence of the topology with results of Jahnke, Simon and Walsberg [Dp-minimal valued fields, J. Symbolic Logic 82(1) (2017) 151–165], it follows that dp-minimal fields that are neither algebraically closed nor real closed admit nontrivial definable Henselian valuations. These results are a key stepping stone toward the classification of dp-minimal fields in [Fun with fields, Ph.D. thesis, University of California, Berkeley (2016)].","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"24 1","pages":"1850007:1-1850007:23"},"PeriodicalIF":0.9,"publicationDate":"2018-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90624378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Interpolative fusions","authors":"Alex Kruckman, Chieu-Minh Tran, Erik Walsberg","doi":"10.1142/S0219061321500100","DOIUrl":"https://doi.org/10.1142/S0219061321500100","url":null,"abstract":"We define the interpolative fusion [Formula: see text] of a family [Formula: see text] of first-order theories over a common reduct [Formula: see text], a notion that generalizes many examples of random or generic structures in the model-theoretic literature. When each [Formula: see text] is model-complete, [Formula: see text] coincides with the model companion of [Formula: see text]. By obtaining sufficient conditions for the existence of [Formula: see text], we develop new tools to show that theories of interest have model companions.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"99 1","pages":"2150010:1-2150010:38"},"PeriodicalIF":0.9,"publicationDate":"2018-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82798124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Iterability for (transfinite) stacks","authors":"Farmer Schlutzenberg","doi":"10.1142/s0219061321500082","DOIUrl":"https://doi.org/10.1142/s0219061321500082","url":null,"abstract":"We establish natural criteria under which normally iterable premice are iterable for stacks of normal trees. Let [Formula: see text] be a regular uncountable cardinal. Let [Formula: see text] and [Formula: see text] be an [Formula: see text]-sound premouse and [Formula: see text] be an [Formula: see text]-iteration strategy for [Formula: see text] (roughly, a normal [Formula: see text]-strategy). We define a natural condensation property for iteration strategies, inflation condensation. We show that if [Formula: see text] has inflation condensation then [Formula: see text] is [Formula: see text]-iterable (roughly, [Formula: see text] is iterable for length [Formula: see text] stacks of normal trees each of length [Formula: see text]), and moreover, we define a specific such strategy [Formula: see text] and a reduction of stacks via [Formula: see text] to normal trees via [Formula: see text]. If [Formula: see text] has the Dodd-Jensen property and [Formula: see text] then [Formula: see text] has inflation condensation. We also apply some of the techniques developed to prove that if [Formula: see text] has strong hull condensation (introduced independently by John Steel), and [Formula: see text] is [Formula: see text]-generic for an [Formula: see text]-cc forcing, then [Formula: see text] extends to an [Formula: see text]-strategy [Formula: see text] for [Formula: see text] with strong hull condensation, in the sense of [Formula: see text]. Moreover, this extension is unique. We deduce that if [Formula: see text] is [Formula: see text]-generic for a ccc forcing then [Formula: see text] and [Formula: see text] have the same [Formula: see text]-sound, [Formula: see text]-iterable premice which project to [Formula: see text].","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"4 1","pages":"2150008:1-2150008:117"},"PeriodicalIF":0.9,"publicationDate":"2018-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79870223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Defining integer-valued functions in rings of continuous definable functions over a topological field","authors":"Luck Darnière, Marcus Tressl","doi":"10.1142/s0219061320500142","DOIUrl":"https://doi.org/10.1142/s0219061320500142","url":null,"abstract":"Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, or an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained in [Formula: see text] and naturally isomorphic to [Formula: see text], such that the ring of functions [Formula: see text] which take values in [Formula: see text] is definable in [Formula: see text].","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"101 1","pages":"2050014:1-2050014:24"},"PeriodicalIF":0.9,"publicationDate":"2018-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85815674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generic expansions by a reduct","authors":"Christian d'Elb'ee","doi":"10.1142/S0219061321500161","DOIUrl":"https://doi.org/10.1142/S0219061321500161","url":null,"abstract":"Consider the expansion [Formula: see text] of a theory [Formula: see text] by a predicate for a submodel of a reduct [Formula: see text] of [Formula: see text]. We present a setup in which this expansion admits a model companion [Formula: see text]. We show that some of the nice features of the theory [Formula: see text] transfer to [Formula: see text]. In particular, we study conditions for which this expansion preserves the [Formula: see text]-ness, the simplicity or the stability of the starting theory [Formula: see text]. We give concrete examples of new [Formula: see text] not simple theories obtained by this process, among them the expansion of a perfect [Formula: see text]-free PAC field of positive characteristic by generic additive subgroups, and the expansion of an algebraically closed field of any characteristic by a generic multiplicative subgroup.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"1 1","pages":"2150016:1-2150016:44"},"PeriodicalIF":0.9,"publicationDate":"2018-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91218926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Karen Bakke Haga, David Schrittesser, Asger Törnquist
{"title":"Maximal almost disjoint families, determinacy, and forcing","authors":"Karen Bakke Haga, David Schrittesser, Asger Törnquist","doi":"10.1142/S0219061321500264","DOIUrl":"https://doi.org/10.1142/S0219061321500264","url":null,"abstract":"We study the notion of [Formula: see text]-MAD families where [Formula: see text] is a Borel ideal on [Formula: see text]. We show that if [Formula: see text] is any finite or countably iterated Fubini product of the ideal of finite sets [Formula: see text], then there are no analytic infinite [Formula: see text]-MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective [Formula: see text]-MAD families; and under the full Axiom of Determinacy + [Formula: see text] or under [Formula: see text] there are no infinite [Formula: see text]-mad families. Similar results are obtained in Solovay’s model. These results apply in particular to the ideal [Formula: see text], which corresponds to the classical notion of MAD families, as well as to the ideal [Formula: see text]. The proofs combine ideas from invariant descriptive set theory and forcing.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"64 1","pages":"2150026:1-2150026:42"},"PeriodicalIF":0.9,"publicationDate":"2018-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80883666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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