Pub Date : 2019-01-04DOI: 10.1142/s0219061320500191
Monroe Eskew
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.
{"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}
Pub Date : 2018-12-27DOI: 10.1142/s021906132250012x
L. Kolodziejczyk, Neil Thapen
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.
{"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}
Pub Date : 2018-12-27DOI: 10.1142/s0219061320500099
E. Baro, Pantelis E. Eleftheriou, Y. Peterzil
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 $G$.
{"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}
Pub Date : 2018-11-20DOI: 10.1142/S0219061318500113
O. Spinas
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.
我们一致地证明了勒贝格零理想不可以土基约化为西尔弗零理想。这与作者Spinas [Silver trees and Cohen realals, Israel J. Math. 211(2016) 473-480]最近的结果形成鲜明对比，后者可归约为Silver理想。
{"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}
Pub Date : 2018-11-20DOI: 10.1142/S0219061318500071
Will Johnson
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)].
在任意非强极小的dp-极小域[公式:见文]上构造了一个非平凡可定义V型域拓扑，并证明了[公式:见文]的可定义子集具有小边界。利用该拓扑及其性质，我们证明了在任意dp-极小域[公式:见文]中，可定义集的dp-rank在族中是可定义变化的，完备型的dp-rank用代数闭包表示，并且[公式:见文]对所有[公式:见文]都是有限的。此外，通过将拓扑的存在性与Jahnke, Simon和Walsberg [dp-极小值域，J. Symbolic Logic 82(1)(2017) 151-165]的结果相结合，可以得出既不是代数闭也不是实闭的dp-极小域承认非平凡可定义的Henselian值。这些结果是[Fun with fields, phd . thesis, University of California, Berkeley(2016)]中dp-minimal field分类的关键垫脚石。
{"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}
Pub Date : 2018-11-14DOI: 10.1142/S0219061321500100
Alex Kruckman, Chieu-Minh Tran, Erik Walsberg
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.
{"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}
Pub Date : 2018-11-09DOI: 10.1142/s0219061321500082
Farmer Schlutzenberg
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].
我们建立了自然准则，在该准则下，正常可迭代的先验对象对于正常树堆栈是可迭代的。设[公式:见正文]为规则不可数基数。假设[公式:参见文本]和[公式:参见文本]是[公式:参见文本]的声音预鼠标，[公式:参见文本]是[公式:参见文本]的[公式:参见文本]的[公式:参见文本]的迭代策略(粗略地说，是一个正常的[公式:参见文本]策略)。我们定义了迭代策略的自然凝聚性质，膨胀凝聚。我们表明，如果[Formula: see text]具有膨胀冷凝，那么[Formula: see text]就是[Formula: see text]-可迭代的(粗略地说，[Formula: see text]对于长度[Formula: see text]的正常树堆栈是可迭代的[Formula: see text])，此外，我们定义了一个特定的策略[Formula: see text]，并通过[Formula: see text]将堆栈减少到通过[Formula: see text]的正常树。如果[公式:见文本]具有多德-詹森属性，而[公式:见文本]则具有膨胀凝结。我们还应用开发的一些技术来证明，如果[公式:见文]具有强船体凝结(由John Steel独立介绍)，并且[公式:见文]是[公式:见文]-通用的[公式:见文]-cc强迫，那么[公式:见文]扩展到[公式:见文]-策略[公式:见文]具有强船体凝结，在[公式:见文]的意义上。此外，这个扩展是唯一的。我们推断，如果[公式:见文]是[公式:见文]- ccc强迫的通用，那么[公式:见文]和[公式:见文]具有相同的[公式:见文]-声音，[公式:见文]-可迭代的前提，投射到[公式:见文]。
{"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}
Pub Date : 2018-10-30DOI: 10.1142/s0219061320500142
Luck Darnière, Marcus Tressl
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].
设[公式:见文本]是有序字段[公式:见文本]或值字段[公式:见文本]的展开。给定一个可定义集合[公式:见文]，设[公式:见文]为从[公式:见文]到[公式:见文]的连续可定义函数环。在[公式:见文]的几何和结构[公式:见文]的非常温和的假设下，特别是当[公式:见文]是[公式:见文]-极小或[公式:见文]-极小，或局部域的展开式时，我们证明了整数环[公式:见文]可解释于[公式:见文]。如果[Formula: see text]是[Formula: see text]的最小值，且[Formula: see text]是纯维度[Formula: see text]的可定义连通，则[Formula: see text]定义了子带[Formula: see text]。如果[公式:见文]是[公式:见文]的极小值，并且[公式:见文]没有孤立点，则存在一个包含在[公式:见文]中的离散环[公式:见文]，并且与[公式:见文]自然同构，使得取[公式:见文]中值的函数[公式:见文]的环在[公式:见文]中可定义。
{"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}
Pub Date : 2018-10-27DOI: 10.1142/S0219061321500161
Christian d'Elb'ee
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.
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Pub Date : 2018-10-06DOI: 10.1142/S0219061321500264
Karen Bakke Haga, David Schrittesser, Asger Törnquist
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.
我们研究了[Formula: see text]-MAD族的概念，其中[Formula: see text]是[Formula: see text]上的Borel理想。我们证明，如果[公式:见文]是有限集合的理想[公式:见文]的任何有限或可数迭代的富比尼积，则不存在解析无限[公式:见文]-MAD族，并且假设射影确定性和依赖选择，不存在无限射影[公式:见文]-MAD族;在完全决定论公理+[公式:见文]或[公式:见文]下，没有无限的[公式:见文]狂族。在Solovay模型中也得到了类似的结果。这些结果特别适用于理想[公式:见文本]，它与MAD家族的经典概念相对应，也适用于理想[公式:见文本]。这些证明结合了不变描述集合论和强迫的思想。
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