Pub Date : 2022-02-05DOI: 10.1142/s0219061322500076
Philipp Lücke
Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle [Formula: see text] introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above by subtleness, and contains all large cardinals that can be characterized through the validity of the principle [Formula: see text] for all classes of structures defined by formulas in a fixed level of the Lévy hierarchy. Moreover, it turns out that no property that can be characterized through this principle can provably imply strong inaccessibility. The proofs of these results rely heavily on the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor’s classical characterization of supercompactness. In addition, we show that several important weak large cardinal properties, like weak inaccessibility, weak Mahloness or weak [Formula: see text]-indescribability, can be canonically characterized through localized versions of the principle [Formula: see text]. Finally, the techniques developed in the proofs of these characterizations also allow us to show that Hamkin’s weakly compact embedding property is equivalent to Lévy’s notion of weak [Formula: see text]-indescribability.
{"title":"Structural reflection, shrewd cardinals and the size of the continuum","authors":"Philipp Lücke","doi":"10.1142/s0219061322500076","DOIUrl":"https://doi.org/10.1142/s0219061322500076","url":null,"abstract":"Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle [Formula: see text] introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above by subtleness, and contains all large cardinals that can be characterized through the validity of the principle [Formula: see text] for all classes of structures defined by formulas in a fixed level of the Lévy hierarchy. Moreover, it turns out that no property that can be characterized through this principle can provably imply strong inaccessibility. The proofs of these results rely heavily on the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor’s classical characterization of supercompactness. In addition, we show that several important weak large cardinal properties, like weak inaccessibility, weak Mahloness or weak [Formula: see text]-indescribability, can be canonically characterized through localized versions of the principle [Formula: see text]. Finally, the techniques developed in the proofs of these characterizations also allow us to show that Hamkin’s weakly compact embedding property is equivalent to Lévy’s notion of weak [Formula: see text]-indescribability.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"7 1","pages":"2250007:1-2250007:43"},"PeriodicalIF":0.9,"publicationDate":"2022-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86636755","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 : 2021-11-02DOI: 10.1142/s0219061322500118
J. Baldwin, S. Shelah
Theorem: There is a complete sentence [Formula: see text] of [Formula: see text] such that [Formula: see text] has maximal models in a set of cardinals [Formula: see text] that is cofinal in the first measurable [Formula: see text] while [Formula: see text] has no maximal models in any [Formula: see text].
{"title":"Maximal models up to the first measurable in ZFC","authors":"J. Baldwin, S. Shelah","doi":"10.1142/s0219061322500118","DOIUrl":"https://doi.org/10.1142/s0219061322500118","url":null,"abstract":"Theorem: There is a complete sentence [Formula: see text] of [Formula: see text] such that [Formula: see text] has maximal models in a set of cardinals [Formula: see text] that is cofinal in the first measurable [Formula: see text] while [Formula: see text] has no maximal models in any [Formula: see text].","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41366898","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 : 2021-10-06DOI: 10.1142/s0219061322500015
F. Pakhomov, J. Walsh
{"title":"Reducing ω-model reflection to iterated syntactic reflection","authors":"F. Pakhomov, J. Walsh","doi":"10.1142/s0219061322500015","DOIUrl":"https://doi.org/10.1142/s0219061322500015","url":null,"abstract":"","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"33 1","pages":"2250001:1-2250001:40"},"PeriodicalIF":0.9,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82426554","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 : 2021-09-19DOI: 10.1142/s0219061324500065
Tom Benhamou, M. Gitik, Yair Hayut
We study which $kappa$-distributive forcing notions of size $kappa$ can be embedded into tree Prikry forcing notions with $kappa$-complete ultrafilters under various large cardinal assumptions. An alternative formulation -- can the filter of dense open subsets of a $kappa$-distributive forcing notion of size $kappa$ be extended to a $kappa$-complete ultrafilter.
{"title":"The variety of projections of a tree-prikry forcing","authors":"Tom Benhamou, M. Gitik, Yair Hayut","doi":"10.1142/s0219061324500065","DOIUrl":"https://doi.org/10.1142/s0219061324500065","url":null,"abstract":"We study which $kappa$-distributive forcing notions of size $kappa$ can be embedded into tree Prikry forcing notions with $kappa$-complete ultrafilters under various large cardinal assumptions. An alternative formulation -- can the filter of dense open subsets of a $kappa$-distributive forcing notion of size $kappa$ be extended to a $kappa$-complete ultrafilter.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44382452","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}
Ali Enayat had asked whether two halves of Disjunctive Correctness (DC) for the compositional truth predicate are conservative over Peano Arithmetic. In this article, we show that the principle"every true disjunction has a true disjunct"is equivalent to bounded induction for the compositional truth predicate and thus it is not conservative. On the other hand, the converse implication"any disjunction with a true disjunct is true"can be conservatively added to PA. The methods introduced here allow us to give a direct nonconservativeness proof for DC.
Ali Enayat问组合真谓词的两半析取正确性(DC)是否在Peano算术上是保守的。在本文中，我们证明了“每个真析取都有一个真析取”的原理等价于组合真谓词的有界归纳法，因此它不是保守的。另一方面，逆向蕴涵“任何有真析取的析取都为真”可以保守地添加到PA中。本文介绍的方法使我们能够给出直流的直接非保守性证明。
{"title":"The two halves of disjunctive correctness","authors":"Cezary Cie'sli'nski, Mateusz Lelyk, Bartosz Wcislo","doi":"10.1142/s021906132250026x","DOIUrl":"https://doi.org/10.1142/s021906132250026x","url":null,"abstract":"Ali Enayat had asked whether two halves of Disjunctive Correctness (DC) for the compositional truth predicate are conservative over Peano Arithmetic. In this article, we show that the principle\"every true disjunction has a true disjunct\"is equivalent to bounded induction for the compositional truth predicate and thus it is not conservative. On the other hand, the converse implication\"any disjunction with a true disjunct is true\"can be conservatively added to PA. The methods introduced here allow us to give a direct nonconservativeness proof for DC.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"475 1","pages":"2250026:1-2250026:28"},"PeriodicalIF":0.9,"publicationDate":"2021-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76997177","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 : 2021-07-01DOI: 10.1142/s0219061322500106
G. Goldberg
This paper explores several topics related to Woodin’s HOD conjecture. We improve the large cardinal hypothesis of Woodin’s HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable ([Formula: see text])-Jónsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD.
{"title":"Strongly Compact Cardinals and Ordinal Definability","authors":"G. Goldberg","doi":"10.1142/s0219061322500106","DOIUrl":"https://doi.org/10.1142/s0219061322500106","url":null,"abstract":"This paper explores several topics related to Woodin’s HOD conjecture. We improve the large cardinal hypothesis of Woodin’s HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable ([Formula: see text])-Jónsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47681429","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 : 2021-06-24DOI: 10.1142/s0219061323500058
D. Hirschfeldt, C. Jockusch, Jr., P. Schupp
The coarse similarity class $[A]$ of $A$ is the set of all $B$ whose symmetric difference with $A$ has asymptotic density 0. There is a natural metric $delta$ on the space $mathcal{S}$ of coarse similarity classes defined by letting $delta([A],[B])$ be the upper density of the symmetric difference of $A$ and $B$. We study the resulting metric space, showing in particular that between any two distinct points there are continuum many geodesic paths. We also study subspaces of the form ${[A] : A in mathcal U}$ where $mathcal U$ is closed under Turing equivalence, and show that there is a tight connection between topological properties of such a space and computability-theoretic properties of $mathcal U$. We then define a distance between Turing degrees based on Hausdorff distance in this metric space. We adapt a proof of Monin to show that the distances between degrees that occur are exactly 0, 1/2, and 1, and study which of these values occur most frequently in the senses of measure and category. We define a degree to be attractive if the class of all degrees at distance 1/2 from it has measure 1, and dispersive otherwise. We study the distribution of attractive and dispersive degrees. We also study some properties of the metric space of Turing degrees under this Hausdorff distance, in particular the question of which countable metric spaces are isometrically embeddable in it, giving a graph-theoretic sufficient condition. We also study the computability-theoretic and reverse-mathematical aspects of a Ramsey-theoretic theorem due to Mycielski, which in particular implies that there is a perfect set whose elements are mutually 1-random, as well as a perfect set whose elements are mutually 1-generic. Finally, we study the completeness of $(mathcal S,delta)$ from the perspectives of computability theory and reverse mathematics.
{"title":"Coarse computability, the density metric, hausdorff distances between turing degrees, perfect trees, and reverse mathematics","authors":"D. Hirschfeldt, C. Jockusch, Jr., P. Schupp","doi":"10.1142/s0219061323500058","DOIUrl":"https://doi.org/10.1142/s0219061323500058","url":null,"abstract":"The coarse similarity class $[A]$ of $A$ is the set of all $B$ whose symmetric difference with $A$ has asymptotic density 0. There is a natural metric $delta$ on the space $mathcal{S}$ of coarse similarity classes defined by letting $delta([A],[B])$ be the upper density of the symmetric difference of $A$ and $B$. We study the resulting metric space, showing in particular that between any two distinct points there are continuum many geodesic paths. We also study subspaces of the form ${[A] : A in mathcal U}$ where $mathcal U$ is closed under Turing equivalence, and show that there is a tight connection between topological properties of such a space and computability-theoretic properties of $mathcal U$. We then define a distance between Turing degrees based on Hausdorff distance in this metric space. We adapt a proof of Monin to show that the distances between degrees that occur are exactly 0, 1/2, and 1, and study which of these values occur most frequently in the senses of measure and category. We define a degree to be attractive if the class of all degrees at distance 1/2 from it has measure 1, and dispersive otherwise. We study the distribution of attractive and dispersive degrees. We also study some properties of the metric space of Turing degrees under this Hausdorff distance, in particular the question of which countable metric spaces are isometrically embeddable in it, giving a graph-theoretic sufficient condition. We also study the computability-theoretic and reverse-mathematical aspects of a Ramsey-theoretic theorem due to Mycielski, which in particular implies that there is a perfect set whose elements are mutually 1-random, as well as a perfect set whose elements are mutually 1-generic. Finally, we study the completeness of $(mathcal S,delta)$ from the perspectives of computability theory and reverse mathematics.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48787723","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 : 2021-06-24DOI: 10.1142/s0219061322500234
H. Towsner, Rose Weisshaar, L. Westrick
We characterize the completely determined Borel subsets of HYP as exactly the omega_1^{ck} subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalban, Solomon&the third author.
{"title":"Borel combinatorics fail in HYP","authors":"H. Towsner, Rose Weisshaar, L. Westrick","doi":"10.1142/s0219061322500234","DOIUrl":"https://doi.org/10.1142/s0219061322500234","url":null,"abstract":"We characterize the completely determined Borel subsets of HYP as exactly the omega_1^{ck} subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalban, Solomon&the third author.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"74 1","pages":"2250023:1-2250023:19"},"PeriodicalIF":0.9,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80975099","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 : 2021-06-07DOI: 10.1142/s0219061322500210
Jinhoo Ahn, J. Kim, Junguk Lee
In this note, we investigate a new model theoretical tree property, called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and $k$-ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples of ATP and NATP, including pure group, pure fields, and valued fields. More precisely, we prove Mekler's construction for groups, Chatzidakis' style criterion for PAC fields, and the AKE-style principle for valued fields preserving NATP. And we give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras.
{"title":"On the antichain tree property","authors":"Jinhoo Ahn, J. Kim, Junguk Lee","doi":"10.1142/s0219061322500210","DOIUrl":"https://doi.org/10.1142/s0219061322500210","url":null,"abstract":"In this note, we investigate a new model theoretical tree property, called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and $k$-ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples of ATP and NATP, including pure group, pure fields, and valued fields. More precisely, we prove Mekler's construction for groups, Chatzidakis' style criterion for PAC fields, and the AKE-style principle for valued fields preserving NATP. And we give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"40 5 1","pages":"2250021:1-2250021:36"},"PeriodicalIF":0.9,"publicationDate":"2021-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79924985","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 : 2021-05-31DOI: 10.1142/S0219061321500288
B. D. Miller
We generalize the [Formula: see text] dichotomy to doubly-indexed sequences of analytic digraphs. Under a mild definability assumption, we use this generalization to characterize the family of Borel actions of tsi Polish groups on Polish spaces that can be decomposed into countably-many Borel actions admitting complete Borel sets that are lacunary with respect to an open neighborhood of the identity. We also show that if the group in question is non-archimedean, then the inexistence of such a decomposition yields a special kind of continuous embedding of [Formula: see text] into the corresponding orbit equivalence relation.
{"title":"A generalization of the 픾0 dichotomy and a strengthening of the 피0ℕ dichotomy","authors":"B. D. Miller","doi":"10.1142/S0219061321500288","DOIUrl":"https://doi.org/10.1142/S0219061321500288","url":null,"abstract":"We generalize the [Formula: see text] dichotomy to doubly-indexed sequences of analytic digraphs. Under a mild definability assumption, we use this generalization to characterize the family of Borel actions of tsi Polish groups on Polish spaces that can be decomposed into countably-many Borel actions admitting complete Borel sets that are lacunary with respect to an open neighborhood of the identity. We also show that if the group in question is non-archimedean, then the inexistence of such a decomposition yields a special kind of continuous embedding of [Formula: see text] into the corresponding orbit equivalence relation.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"80 1","pages":"2150028:1-2150028:19"},"PeriodicalIF":0.9,"publicationDate":"2021-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80366992","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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