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Model theory of differential fields with finite group actions 有限群作用下微分场的模型理论
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-12-28 DOI: 10.1142/s0219061322500027
D. Hoffmann, Omar Le'on S'anchez
{"title":"Model theory of differential fields with finite group actions","authors":"D. Hoffmann, Omar Le'on S'anchez","doi":"10.1142/s0219061322500027","DOIUrl":"https://doi.org/10.1142/s0219061322500027","url":null,"abstract":"Let G be a finite group. We explore the model theoretic properties of the class of differential fields of characteristic zero in m commuting derivations equipped with a G-action by differential field automorphisms. In the language of G-differential rings (i.e. the language of rings with added symbols for derivations and automorphisms), we prove that this class has a modelcompanion – denoted G -DCF0,m. We then deploy the model-theoretic tools developed in the first author’s paper [11] to show that any model of G -DCF0,m is supersimple (but unstable whenG is nontrivial), a PAC-differential field (and hence differentially large in the sense of the second author and Tressl [30]), and admits elimination of imaginaries after adding a tuple of parameters. We also address model-completeness and supersimplicity of theories of bounded PACdifferential fields (extending the results of Chatzidakis-Pillay [5] on bounded PAC-fields).","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"12 1","pages":"2250002:1-2250002:31"},"PeriodicalIF":0.9,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82173119","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}
引用次数: 1
Extender-based forcings with overlapping extenders and negations of the Shelah Weak Hypothesis 具有重叠扩展器的基于扩展器的强迫与希拉弱假设的否定
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-12-01 DOI: 10.1142/s0219061320500130
M. Gitik
{"title":"Extender-based forcings with overlapping extenders and negations of the Shelah Weak Hypothesis","authors":"M. Gitik","doi":"10.1142/s0219061320500130","DOIUrl":"https://doi.org/10.1142/s0219061320500130","url":null,"abstract":"Extender based Prikry-Magidor forcing for overlapping extenders is introduced. As an application, models with strong forms of negations of the Shelah Weak Hypothesis for various cofinalities are constructed.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"5 1","pages":"2050013:1-2050013:34"},"PeriodicalIF":0.9,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76464539","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}
引用次数: 3
The property "arithmetic-is-recursive" on a cone 圆锥上“算术-递归”的性质
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-11-20 DOI: 10.1142/s0219061321500215
U. Andrews, M. Harrison-Trainor, N. Schweber
{"title":"The property \"arithmetic-is-recursive\" on a cone","authors":"U. Andrews, M. Harrison-Trainor, N. Schweber","doi":"10.1142/s0219061321500215","DOIUrl":"https://doi.org/10.1142/s0219061321500215","url":null,"abstract":"We say that a theory [Formula: see text] satisfies arithmetic-is-recursive if any [Formula: see text]-computable model of [Formula: see text] has an [Formula: see text]-computable copy; that is, the models of [Formula: see text] satisfy a sort of jump inversion. We give an example of a theory satisfying arithmetic-is-recursive non-trivially and prove that the theories satisfying arithmetic-is-recursive on a cone are exactly those theories with countably many [Formula: see text]-back-and-forth types.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"35 1","pages":"2150021:1-2150021:18"},"PeriodicalIF":0.9,"publicationDate":"2020-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73636152","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}
引用次数: 0
The Ultrapower Axiom and the GCH 超功率公理和GCH
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-10-27 DOI: 10.1142/s0219061321500173
G. Goldberg
{"title":"The Ultrapower Axiom and the GCH","authors":"G. Goldberg","doi":"10.1142/s0219061321500173","DOIUrl":"https://doi.org/10.1142/s0219061321500173","url":null,"abstract":"The Ultrapower Axiom is an abstract combinatorial principle inspired by the fine structure of canonical inner models of large cardinal axioms. In this paper, it is established that the Ultrapower Axiom implies that the Generalized Continuum Hypothesis holds above the least supercompact cardinal.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"30 1","pages":"2150017:1-2150017:22"},"PeriodicalIF":0.9,"publicationDate":"2020-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83863345","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}
引用次数: 1
An unpublished theorem of Solovay on OD partitions of reals into two non-OD parts, revisited 索洛维关于实数的OD划分为两个非OD部分的一个未发表的定理
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-10-07 DOI: 10.1142/s0219061321500148
A. Enayat, V. Kanovei
{"title":"An unpublished theorem of Solovay on OD partitions of reals into two non-OD parts, revisited","authors":"A. Enayat, V. Kanovei","doi":"10.1142/s0219061321500148","DOIUrl":"https://doi.org/10.1142/s0219061321500148","url":null,"abstract":"A definable pair of disjoint non-OD sets of reals (hence, indiscernible sets) exists in the Sacks and [Formula: see text]o-large generic extensions of the constructible universe L. More specifically, if [Formula: see text] is either Sacks generic or [Formula: see text]o generic real over L, then it is true in L[Formula: see text] that there is a lightface [Formula: see text] equivalence relation Q on the [Formula: see text] set [Formula: see text] with exactly two equivalence classes, and both those classes are non-OD sets.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"54 1","pages":"2150014:1-2150014:22"},"PeriodicalIF":0.9,"publicationDate":"2020-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91082252","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}
引用次数: 9
Model-theoretic Elekes-Szabó in the strongly minimal case 模型理论Elekes-Szabó在强极小情况下
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-10-06 DOI: 10.1142/S0219061321500045
A. Chernikov, S. Starchenko
{"title":"Model-theoretic Elekes-Szabó in the strongly minimal case","authors":"A. Chernikov, S. Starchenko","doi":"10.1142/S0219061321500045","DOIUrl":"https://doi.org/10.1142/S0219061321500045","url":null,"abstract":"We prove a generalization of the Elekes–Szabó theorem [G. Elekes and E. Szabó, How to find groups? (and how to use them in Erdos geometry?), Combinatorica 32(5) 537–571 (2012)] for relations definable in strongly minimal structures that are interpretable in distal structures.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"17 1","pages":"2150004:1-2150004:20"},"PeriodicalIF":0.9,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81174572","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}
引用次数: 9
Compactness versus hugeness at successor cardinals 后继基数的紧凑性与巨大性
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-09-29 DOI: 10.1142/s0219061322500167
Sean D. Cox, Monroe Eskew
{"title":"Compactness versus hugeness at successor cardinals","authors":"Sean D. Cox, Monroe Eskew","doi":"10.1142/s0219061322500167","DOIUrl":"https://doi.org/10.1142/s0219061322500167","url":null,"abstract":"If $kappa$ is regular and $2^{<kappa}leqkappa^+$, then the existence of a weakly presaturated ideal on $kappa^+$ implies $square^*_kappa$. This partially answers a question of Foreman and Magidor about the approachability ideal on $omega_2$. As a corollary, we show that if there is a presaturated ideal $I$ on $omega_2$ such that $mathcal{P}(omega_2)/I$ is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"646 1","pages":"2250016:1-2250016:16"},"PeriodicalIF":0.9,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77536849","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}
引用次数: 1
Exact saturation in pseudo-elementary classes for simple and stable theories 简单稳定理论的准初等类的精确饱和
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-09-17 DOI: 10.1142/s0219061322500209
Itay Kaplan, N. Ramsey, S. Shelah
{"title":"Exact saturation in pseudo-elementary classes for simple and stable theories","authors":"Itay Kaplan, N. Ramsey, S. Shelah","doi":"10.1142/s0219061322500209","DOIUrl":"https://doi.org/10.1142/s0219061322500209","url":null,"abstract":"We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals, satisfying mild set-theoretic hypotheses, which had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analogue of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"41 1","pages":"2250020:1-2250020:26"},"PeriodicalIF":0.9,"publicationDate":"2020-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77126301","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}
引用次数: 0
Canonical fragments of the strong reflection principle 强反射原理的规范片段
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-09-13 DOI: 10.1142/S0219061321500239
G. Fuchs
{"title":"Canonical fragments of the strong reflection principle","authors":"G. Fuchs","doi":"10.1142/S0219061321500239","DOIUrl":"https://doi.org/10.1142/S0219061321500239","url":null,"abstract":"For an arbitrary forcing class [Formula: see text], the [Formula: see text]-fragment of Todorčević’s strong reflection principle SRP is isolated in such a way that (1) the forcing axiom for [Formula: see text] implies the [Formula: see text]-fragment of SRP , (2) the stationary set preserving fragment of SRP is the full principle SRP , and (3) the subcomplete fragment of SRP implies the major consequences of the subcomplete forcing axiom. This fragment of SRP is consistent with CH , and even with Jensen’s principle [Formula: see text]. Along the way, some hitherto unknown effects of (the subcomplete fragment of) SRP on mutual stationarity are explored, and some limitations to the extent to which fragments of SRP may capture the effects of their corresponding forcing axioms are established.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"26 1","pages":"2150023:1-2150023:58"},"PeriodicalIF":0.9,"publicationDate":"2020-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78247006","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}
引用次数: 4
New jump operators on equivalence relations 等价关系上的新跳转算子
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-08-15 DOI: 10.1142/S0219061322500155
J. Clemens, Samuel Coskey
{"title":"New jump operators on equivalence relations","authors":"J. Clemens, Samuel Coskey","doi":"10.1142/S0219061322500155","DOIUrl":"https://doi.org/10.1142/S0219061322500155","url":null,"abstract":"We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group $Gamma$ we introduce the $Gamma$-jump. We study the elementary properties of the $Gamma$-jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups $Gamma$, the $Gamma$-jump is emph{proper} in the sense that for any Borel equivalence relation $E$ the $Gamma$-jump of $E$ is strictly higher than $E$ in the Borel reducibility hierarchy. On the other hand there are examples of groups $Gamma$ for which the $Gamma$-jump is not proper. To establish properness, we produce an analysis of Borel equivalence relations induced by continuous actions of the automorphism group of what we denote the infinite $Gamma$-tree, and relate these to iterates of the $Gamma$-jump. We also produce several new examples of equivalence relations that arise from applying the $Gamma$-jump to classically studied equivalence relations and derive generic ergodicity results related to these. We apply our results to show that the complexity of the isomorphism problem for countable scattered linear orders properly increases with the rank.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"34 12 1","pages":"2250015:1-2250015:44"},"PeriodicalIF":0.9,"publicationDate":"2020-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77918601","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}
引用次数: 6
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