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":"1 1","pages":""},"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":" ","pages":""},"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":"105 1","pages":"2250008:1-2250008:38"},"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":"30 1","pages":"2250005:1-2250005:41"},"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":"73 1","pages":"2250022:1-2250022:18"},"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":" ","pages":""},"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}
Pub Date : 2022-03-11DOI: 10.1142/s0219061323500022
David J. Fern'andez-Bret'on
In the context of $mathsf{ZF}$, we analyze a version of Hindman's finite unions theorem on infinite sets, which normally requires the Axiom of Choice to be proved. We establish the implication relations between this statement and various classical weak choice principles, thus precisely locating the strength of the statement as a weak form of the $mathsf{AC}$.
{"title":"Hindman's theorem in the hierarchy of choice principles","authors":"David J. Fern'andez-Bret'on","doi":"10.1142/s0219061323500022","DOIUrl":"https://doi.org/10.1142/s0219061323500022","url":null,"abstract":"In the context of $mathsf{ZF}$, we analyze a version of Hindman's finite unions theorem on infinite sets, which normally requires the Axiom of Choice to be proved. We establish the implication relations between this statement and various classical weak choice principles, thus precisely locating the strength of the statement as a weak form of the $mathsf{AC}$.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45407190","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 : 2022-03-04DOI: 10.1142/s0219061323500101
J. Zapletal
If [Formula: see text] is a closed Noetherian graph on a [Formula: see text]-compact Polish space with no infinite cliques, it is consistent with the choiceless set theory ZF[Formula: see text][Formula: see text][Formula: see text]DC that [Formula: see text] is countably chromatic and there is no Vitali set.
{"title":"Coloring closed Noetherian graphs","authors":"J. Zapletal","doi":"10.1142/s0219061323500101","DOIUrl":"https://doi.org/10.1142/s0219061323500101","url":null,"abstract":"If [Formula: see text] is a closed Noetherian graph on a [Formula: see text]-compact Polish space with no infinite cliques, it is consistent with the choiceless set theory ZF[Formula: see text][Formula: see text][Formula: see text]DC that [Formula: see text] is countably chromatic and there is no Vitali set.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47044052","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 : 2022-03-04DOI: 10.1142/s0219061324500053
Konstantinos Kartas
When $k$ is a finite field, Becker-Denef-Lipschitz (1979) observed that the total residue map $text{res}:k(!(t)!)to k$, which picks out the constant term of the Laurent series, is definable in the language of rings with a parameter for $t$. Driven by this observation, we study the theory $text{VF}_{text{res},iota}$ of valued fields equipped with a linear form $text{res}:Kto k$ which specializes to the residue map on the valuation ring. We prove that $text{VF}_{text{res},iota}$ does not admit a model companion. In addition, we show that the power series field $(k(!(t)!),text{res})$, equipped with such a total residue map, is undecidable whenever $k$ is an infinite field. As a consequence, we get that $(mathbb{C}(!(t)!), text{Res}_0)$ is undecidable, where $text{Res}_0:mathbb{C}(!(t)!)to mathbb{C}:fmapsto text{Res}_0(f)$ maps $f$ to its complex residue at $0$.
{"title":"Valued fields with a total residue map","authors":"Konstantinos Kartas","doi":"10.1142/s0219061324500053","DOIUrl":"https://doi.org/10.1142/s0219061324500053","url":null,"abstract":"When $k$ is a finite field, Becker-Denef-Lipschitz (1979) observed that the total residue map $text{res}:k(!(t)!)to k$, which picks out the constant term of the Laurent series, is definable in the language of rings with a parameter for $t$. Driven by this observation, we study the theory $text{VF}_{text{res},iota}$ of valued fields equipped with a linear form $text{res}:Kto k$ which specializes to the residue map on the valuation ring. We prove that $text{VF}_{text{res},iota}$ does not admit a model companion. In addition, we show that the power series field $(k(!(t)!),text{res})$, equipped with such a total residue map, is undecidable whenever $k$ is an infinite field. As a consequence, we get that $(mathbb{C}(!(t)!), text{Res}_0)$ is undecidable, where $text{Res}_0:mathbb{C}(!(t)!)to mathbb{C}:fmapsto text{Res}_0(f)$ maps $f$ to its complex residue at $0$.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"13 4","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41293526","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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