Pub Date : 2023-02-10DOI: 10.1142/s0219061323500046
W. Mitchell, Ernest Schimmerling
{"title":"Covering at limit cardinals of K","authors":"W. Mitchell, Ernest Schimmerling","doi":"10.1142/s0219061323500046","DOIUrl":"https://doi.org/10.1142/s0219061323500046","url":null,"abstract":"","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48117338","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 : 2023-01-01DOI: 10.1142/S0219061322500313
J. Freitag, Rémi Jaoui, Rahim Moosa
. It is shown that if p ∈ S ( A ) is a complete type of Lascar rank at least 2, in the theory of differentially closed fields of characteristic zero, then there exists a pair of realisations a 1 ,a 2 such that p has a nonalgebraic forking extension over Aa 1 a 2 . Moreover, if A is contained in the field of constants then p already has a nonalgebraic forking extension over Aa 1 . The results are also formulated in a more general setting.
． 证明了在特征为0的微分闭域理论中，如果p∈S (A)是至少为2的完备型Lascar秩，则存在一对实现A 1, A 2，使得p在A 1 A 2上具有非代数分叉扩展。此外，如果A包含在常数域中，则p在aa1上已经具有非代数分叉扩展。结果也在更一般的情况下制定。
{"title":"The degree of nonminimality is at most 2","authors":"J. Freitag, Rémi Jaoui, Rahim Moosa","doi":"10.1142/S0219061322500313","DOIUrl":"https://doi.org/10.1142/S0219061322500313","url":null,"abstract":". It is shown that if p ∈ S ( A ) is a complete type of Lascar rank at least 2, in the theory of differentially closed fields of characteristic zero, then there exists a pair of realisations a 1 ,a 2 such that p has a nonalgebraic forking extension over Aa 1 a 2 . Moreover, if A is contained in the field of constants then p already has a nonalgebraic forking extension over Aa 1 . The results are also formulated in a more general setting.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80276776","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 : 2023-01-01DOI: 10.1142/S0219061322500295
William Chan, Stephen Jackson, Nam Trang
Assume ZF+AD. The following two continuity results for functions on certain subsets of P(ω1) and P(ω2) will be shown: For every < ω1 and function Φ : [ω1] → ω1, there is a club C ⊆ ω1 and a ζ < so that for all f, g ∈ [C] ∗, if f ζ = g ζ and sup(f) = sup(g), then Φ(f) = Φ(g). For every < ω2 and function Φ : [ω2] → ω2, there is an ω-club C ⊆ ω2 and a ζ < so that for all f, g ∈ [C] ∗, if f ζ = g ζ and sup(f) = sup(g), then Φ(f) = Φ(g). The previous two continuity results will be used to distinguish cardinals below P(ω2): |[ω1] | < |[ω1]1 |. |[ω2] | < |ω2]1 | < |[ω2]1 | < |[ω2]2 |. ¬(|[ω1]1 | ≤ [ω2] |). ¬(|[ω1]1 | ≤ ([ω2]1 |). [ω1] has the Jónsson property: That is, for every Φ : ([ω1]) → [ω1] , there is an X ⊆ [ω1] with |X| = |[ω1] | so that Φ[
{"title":"More definable combinatorics around the first and second uncountable cardinals","authors":"William Chan, Stephen Jackson, Nam Trang","doi":"10.1142/S0219061322500295","DOIUrl":"https://doi.org/10.1142/S0219061322500295","url":null,"abstract":"Assume ZF+AD. The following two continuity results for functions on certain subsets of P(ω1) and P(ω2) will be shown: For every < ω1 and function Φ : [ω1] → ω1, there is a club C ⊆ ω1 and a ζ < so that for all f, g ∈ [C] ∗, if f ζ = g ζ and sup(f) = sup(g), then Φ(f) = Φ(g). For every < ω2 and function Φ : [ω2] → ω2, there is an ω-club C ⊆ ω2 and a ζ < so that for all f, g ∈ [C] ∗, if f ζ = g ζ and sup(f) = sup(g), then Φ(f) = Φ(g). The previous two continuity results will be used to distinguish cardinals below P(ω2): |[ω1] | < |[ω1]1 |. |[ω2] | < |ω2]1 | < |[ω2]1 | < |[ω2]2 |. ¬(|[ω1]1 | ≤ [ω2] |). ¬(|[ω1]1 | ≤ ([ω2]1 |). [ω1] has the Jónsson property: That is, for every Φ : ([ω1]) → [ω1] , there is an X ⊆ [ω1] with |X| = |[ω1] | so that Φ[","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82602286","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 : 2022-12-17DOI: 10.1142/s0219061322500271
A. Rodriguez Fanlo
The aim of this paper is to generalize and improve two of the main model-theoretic results of “Stable group theory and approximate subgroups” by Hrushovski to the context of piecewise hyperdefinable sets. The first one is the existence of Lie models. The second one is the Stabilizer Theorem. In the process, a systematic study of the structure of piecewise hyperdefinable sets is developed. In particular, we show the most significant properties of their logic topologies.
{"title":"On piecewise hyperdefinable groups","authors":"A. Rodriguez Fanlo","doi":"10.1142/s0219061322500271","DOIUrl":"https://doi.org/10.1142/s0219061322500271","url":null,"abstract":"The aim of this paper is to generalize and improve two of the main model-theoretic results of “Stable group theory and approximate subgroups” by Hrushovski to the context of piecewise hyperdefinable sets. The first one is the existence of Lie models. The second one is the Stabilizer Theorem. In the process, a systematic study of the structure of piecewise hyperdefinable sets is developed. In particular, we show the most significant properties of their logic topologies.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64341575","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 : 2022-09-09DOI: 10.1142/s0219061324500028
Barbara F. Csima, D. Rossegger
We give a characterization of the strong degrees of categoricity of computable structures greater or equal to $mathbf 0''$. They are precisely the emph{treeable} degrees -- the least degrees of paths through computable trees -- that compute $mathbf 0''$. As a corollary, we obtain several new examples of degrees of categoricity. Among them we show that every degree $mathbf d$ with $mathbf 0^{(alpha)}leq mathbf dleq mathbf 0^{(alpha+1)}$ for $alpha$ a computable ordinal greater than $2$ is the strong degree of categoricity of a rigid structure. Using quite different techniques we show that every degree $mathbf d$ with $mathbf 0'leq mathbf dleq mathbf 0''$ is the strong degree of categoricity of a structure. Together with the above example this answers a question of Csima and Ng. To complete the picture we show that there is a degree $mathbf d$ with $mathbf 0'
{"title":"Degrees of categoricity and treeable degrees","authors":"Barbara F. Csima, D. Rossegger","doi":"10.1142/s0219061324500028","DOIUrl":"https://doi.org/10.1142/s0219061324500028","url":null,"abstract":"We give a characterization of the strong degrees of categoricity of computable structures greater or equal to $mathbf 0''$. They are precisely the emph{treeable} degrees -- the least degrees of paths through computable trees -- that compute $mathbf 0''$. As a corollary, we obtain several new examples of degrees of categoricity. Among them we show that every degree $mathbf d$ with $mathbf 0^{(alpha)}leq mathbf dleq mathbf 0^{(alpha+1)}$ for $alpha$ a computable ordinal greater than $2$ is the strong degree of categoricity of a rigid structure. Using quite different techniques we show that every degree $mathbf d$ with $mathbf 0'leq mathbf dleq mathbf 0''$ is the strong degree of categoricity of a structure. Together with the above example this answers a question of Csima and Ng. To complete the picture we show that there is a degree $mathbf d$ with $mathbf 0'<mathbf d<mathbf 0''$ that is not the degree of categoricity of a rigid structure.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43179565","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 : 2022-06-22DOI: 10.1142/s0219061322500088
Stefan Hoffelner
We generically construct a model in which the [Formula: see text]-separation property is true, i.e. every pair of disjoint [Formula: see text]-sets can be separated by a [Formula: see text]-definable set. This answers an old question from the problem list “Surrealist landscape with figures” by A. Mathias from 1968. We also construct a model in which the (lightface) [Formula: see text]-separation property is true.
{"title":"Forcing the Σ31-separation property","authors":"Stefan Hoffelner","doi":"10.1142/s0219061322500088","DOIUrl":"https://doi.org/10.1142/s0219061322500088","url":null,"abstract":"We generically construct a model in which the [Formula: see text]-separation property is true, i.e. every pair of disjoint [Formula: see text]-sets can be separated by a [Formula: see text]-definable set. This answers an old question from the problem list “Surrealist landscape with figures” by A. Mathias from 1968. We also construct a model in which the (lightface) [Formula: see text]-separation property is true.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78558140","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 : 2022-06-22DOI: 10.1142/s0219061322500052
G. Goldberg
The Ultrapower Axiom is a combinatorial principle concerning the structure of large cardinals that is true in all known canonical inner models of set theory. A longstanding test question for inner model theory is the equiconsistency of strongly compact and supercompact cardinals. In this paper, it is shown that under the Ultrapower Axiom, the least strongly compact cardinal is supercompact. A number of stronger results are established, setting the stage for a complete analysis of strong compactness and supercompactness under UA that will be carried out in the sequel to this paper.
{"title":"Strong compactness and the ultrapower axiom I: the least strongly compact cardinal","authors":"G. Goldberg","doi":"10.1142/s0219061322500052","DOIUrl":"https://doi.org/10.1142/s0219061322500052","url":null,"abstract":"The Ultrapower Axiom is a combinatorial principle concerning the structure of large cardinals that is true in all known canonical inner models of set theory. A longstanding test question for inner model theory is the equiconsistency of strongly compact and supercompact cardinals. In this paper, it is shown that under the Ultrapower Axiom, the least strongly compact cardinal is supercompact. A number of stronger results are established, setting the stage for a complete analysis of strong compactness and supercompactness under UA that will be carried out in the sequel to this paper.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84381353","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 : 2022-06-17DOI: 10.1142/s0219061322500222
Barbara F. Csima, K. Ng
{"title":"Every Δ20 degree is a strong degree of categoricity","authors":"Barbara F. Csima, K. Ng","doi":"10.1142/s0219061322500222","DOIUrl":"https://doi.org/10.1142/s0219061322500222","url":null,"abstract":"","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83566442","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 : 2022-05-03DOI: 10.1142/s0219061323500083
Nicholas Pischke
Accretive and monotone operator theory are central branches of nonlinear functional analysis and constitute the abstract study of set-valued mappings between function spaces. This paper deals with the computational properties of certain large classes of operators, namely accretive and (generalized) monotone set-valued ones. In particular, we develop (and extend) for this field the theoretical framework of proof mining, a program in mathematical logic that seeks to extract computational information from prima facie `non-computational' proofs from the mainstream literature. To this end, we establish logical metatheorems that guarantee and quantify the computational content of theorems pertaining to accretive and (generalized) monotone set-valued operators. On one hand, our results unify a number of recent case studies, while they also provide characterizations of central analytical notions in terms of proof theoretic ones on the other, which provides a crucial perspective on needed quantitative assumptions in future applications of proof mining to these branches.
{"title":"Logical Metatheorems for Accretive and (Generalized) Monotone Set-Valued Operators","authors":"Nicholas Pischke","doi":"10.1142/s0219061323500083","DOIUrl":"https://doi.org/10.1142/s0219061323500083","url":null,"abstract":"Accretive and monotone operator theory are central branches of nonlinear functional analysis and constitute the abstract study of set-valued mappings between function spaces. This paper deals with the computational properties of certain large classes of operators, namely accretive and (generalized) monotone set-valued ones. In particular, we develop (and extend) for this field the theoretical framework of proof mining, a program in mathematical logic that seeks to extract computational information from prima facie `non-computational' proofs from the mainstream literature. To this end, we establish logical metatheorems that guarantee and quantify the computational content of theorems pertaining to accretive and (generalized) monotone set-valued operators. On one hand, our results unify a number of recent case studies, while they also provide characterizations of central analytical notions in terms of proof theoretic ones on the other, which provides a crucial perspective on needed quantitative assumptions in future applications of proof mining to these branches.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46950370","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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