Pub Date : 2021-02-12DOI: 10.1142/s0219061322500192
J. Bergfalk, M. Hrusák, C. Lambie-Hanson
A question dating to Sibe Mardev{s}i'{c} and Andrei Prasolov's 1988 work Strong homology is not additive, and motivating a considerable amount of set theoretic work in the ensuing years, is that of whether it is consistent with the ZFC axioms for the higher derived limits $mathrm{lim}^n$ $(n>0)$ of a certain inverse system $mathbf{A}$ indexed by ${^omega}omega$ to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all $n$-coherent families of functions indexed by ${^omega}omega$ to be trivial. In this paper, we prove that, in any forcing extension given by adjoining $beth_omega$-many Cohen reals, $mathrm{lim}^n mathbf{A}$ vanishes for all $n>0$. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher dimensional $Delta$-system lemmas. This work removes all large cardinal hypotheses from the main result of arXiv:1907.11744 and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of $mathrm{lim}^n mathbf{A}$ for all $n>0$.
Sibe Marde v{s}和Andrei Prasolov在1988年的著作《强同调不是可加的》(Strong homology is not additive)中提出的一个问题是，对于以${^omega}omega$为索引的某逆系统$mathbf{A}$的较高推导极限$mathrm{lim}^n$$(n>0)$是否与ZFC公理相一致，这个问题在随后的几年里激发了大量的集合论工作。这个问题的一个等价的表述是，是否所有$n$ -相干族的函数都以${^omega}omega$为索引是平凡的。在本文中，我们证明了在任意由相邻的$beth_omega$ -多个Cohen实数给出的强迫扩展中，对于所有$n>0$, $mathrm{lim}^n mathbf{A}$都消失。我们的证明包括对高维$Delta$ -系统引理的强迫扩展和重复应用的详细组合分析。这项工作从arXiv:1907.11744的主要结果中删除了所有大的基本假设，并大大降低了已知与所有$n>0$的$mathrm{lim}^n mathbf{A}$同时消失相容的连续统的最小值。
{"title":"Simultaneously vanishing higher derived limits without large cardinals","authors":"J. Bergfalk, M. Hrusák, C. Lambie-Hanson","doi":"10.1142/s0219061322500192","DOIUrl":"https://doi.org/10.1142/s0219061322500192","url":null,"abstract":"A question dating to Sibe Mardev{s}i'{c} and Andrei Prasolov's 1988 work Strong homology is not additive, and motivating a considerable amount of set theoretic work in the ensuing years, is that of whether it is consistent with the ZFC axioms for the higher derived limits $mathrm{lim}^n$ $(n>0)$ of a certain inverse system $mathbf{A}$ indexed by ${^omega}omega$ to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all $n$-coherent families of functions indexed by ${^omega}omega$ to be trivial. In this paper, we prove that, in any forcing extension given by adjoining $beth_omega$-many Cohen reals, $mathrm{lim}^n mathbf{A}$ vanishes for all $n>0$. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher dimensional $Delta$-system lemmas. This work removes all large cardinal hypotheses from the main result of arXiv:1907.11744 and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of $mathrm{lim}^n mathbf{A}$ for all $n>0$.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82652032","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 : 2021-01-08DOI: 10.1142/s0219061322500064
D. Asperó, M. Viale
We introduce bounded category forcing axioms for well-behaved classes [Formula: see text]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [Formula: see text] modulo forcing in [Formula: see text], for some cardinal [Formula: see text] naturally associated to [Formula: see text]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [Formula: see text] — to classes [Formula: see text] with [Formula: see text]. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms but can be forced under mild large cardinal assumptions on [Formula: see text]. We also show the existence of many classes [Formula: see text] with [Formula: see text] giving rise to pairwise incompatible theories for [Formula: see text].
{"title":"Incompatible bounded category forcing axioms","authors":"D. Asperó, M. Viale","doi":"10.1142/s0219061322500064","DOIUrl":"https://doi.org/10.1142/s0219061322500064","url":null,"abstract":"We introduce bounded category forcing axioms for well-behaved classes [Formula: see text]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [Formula: see text] modulo forcing in [Formula: see text], for some cardinal [Formula: see text] naturally associated to [Formula: see text]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [Formula: see text] — to classes [Formula: see text] with [Formula: see text]. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms but can be forced under mild large cardinal assumptions on [Formula: see text]. We also show the existence of many classes [Formula: see text] with [Formula: see text] giving rise to pairwise incompatible theories for [Formula: see text].","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86208738","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 : 2021-01-01DOI: 10.1142/s021906132192001x
L. Barto, M. Kompatscher, M. Olsák, Trung Van Pham, M. Pinsker
{"title":"Erratum: Equations in oligomorphic clones and the constraint satisfaction problem for ω-categorical structures","authors":"L. Barto, M. Kompatscher, M. Olsák, Trung Van Pham, M. Pinsker","doi":"10.1142/s021906132192001x","DOIUrl":"https://doi.org/10.1142/s021906132192001x","url":null,"abstract":"","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90356660","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 : 2020-12-28DOI: 10.1142/s0219061322500027
D. Hoffmann, Omar Le'on S'anchez
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).
{"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":null,"pages":null},"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}
Pub Date : 2020-12-01DOI: 10.1142/s0219061320500130
M. Gitik
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.
{"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":null,"pages":null},"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}
Pub Date : 2020-11-20DOI: 10.1142/s0219061321500215
U. Andrews, M. Harrison-Trainor, N. Schweber
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.
{"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":null,"pages":null},"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}
Pub Date : 2020-10-27DOI: 10.1142/s0219061321500173
G. Goldberg
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.
{"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":null,"pages":null},"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}
Pub Date : 2020-10-07DOI: 10.1142/s0219061321500148
A. Enayat, V. Kanovei
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.
{"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":null,"pages":null},"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}
Pub Date : 2020-10-06DOI: 10.1142/S0219061321500045
A. Chernikov, S. Starchenko
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.
{"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":null,"pages":null},"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}
Pub Date : 2020-09-29DOI: 10.1142/s0219061322500167
Sean D. Cox, Monroe Eskew
If $kappa$ is regular and $2^{
如果$kappa$是正则的并且$2^{
{"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":null,"pages":null},"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}
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