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":"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}
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":"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}
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":"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}
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":"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}
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":"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}
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":"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}
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":"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}
Pub Date : 2020-09-17DOI: 10.1142/s0219061322500209
Itay Kaplan, N. Ramsey, S. Shelah
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.
{"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}
Pub Date : 2020-09-13DOI: 10.1142/S0219061321500239
G. Fuchs
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.
{"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}
Pub Date : 2020-08-15DOI: 10.1142/S0219061322500155
J. Clemens, Samuel Coskey
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.
{"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}
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