We study various combinatorial properties, and the implications between them, for filters generated by infinite-dimensional subspaces of a countable vector space. These properties are analogous to selectivity for ultrafilters on the natural numbers and stability for ordered-union ultrafilters on FIN.
{"title":"Filters on a countable vector space","authors":"Iian B. Smythe","doi":"10.4064/fm197-5-2022","DOIUrl":"https://doi.org/10.4064/fm197-5-2022","url":null,"abstract":"We study various combinatorial properties, and the implications between them, for filters generated by infinite-dimensional subspaces of a countable vector space. These properties are analogous to selectivity for ultrafilters on the natural numbers and stability for ordered-union ultrafilters on FIN.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49076452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. For every infinite (countable discrete) amenable group G and every positive integer d we construct a minimal G -action of mean dimension d/ 2 which cannot be embedded in the full G -shift on ([0 , 1] d ) G .
{"title":"Mean dimension and a non-embeddable example\u0000for amenable group actions","authors":"Lei Jin, K. Park, Yixiao Qiao","doi":"10.4064/fm921-1-2021","DOIUrl":"https://doi.org/10.4064/fm921-1-2021","url":null,"abstract":". For every infinite (countable discrete) amenable group G and every positive integer d we construct a minimal G -action of mean dimension d/ 2 which cannot be embedded in the full G -shift on ([0 , 1] d ) G .","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45184918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the Boolean algebras approach to class forcing can be carried out in the theory Kelley-Morse plus the Choice Scheme (KM + CC) using hyperclass Boolean completions of class partial orders. Applying the Boolean algebras approach, we show that every intermediate model between a model V |= KM + CC and one of its class forcing extensions is itself a class forcing extension if and only if it is simple generated by the classes of V together with a single new class. We show that a model of KM + CC and its class forcing extension may have non-simple intermediate models, and thus, the Intermediate Model Theorem can fail for models of KM + CC.
{"title":"Boolean-valued class forcing","authors":"C. Antos, S. Friedman, V. Gitman","doi":"10.4064/fm20-7-2021","DOIUrl":"https://doi.org/10.4064/fm20-7-2021","url":null,"abstract":"We show that the Boolean algebras approach to class forcing can be carried out in the theory Kelley-Morse plus the Choice Scheme (KM + CC) using hyperclass Boolean completions of class partial orders. Applying the Boolean algebras approach, we show that every intermediate model between a model V |= KM + CC and one of its class forcing extensions is itself a class forcing extension if and only if it is simple generated by the classes of V together with a single new class. We show that a model of KM + CC and its class forcing extension may have non-simple intermediate models, and thus, the Intermediate Model Theorem can fail for models of KM + CC.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70253173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. The Wholeness Axiom (WA) is an axiom schema that asserts the existence of a nontrivial elementary embedding from V to itself. The schema is formulated in the language {∈ , j } , where j is a unary function symbol intended to stand for the embedding. WA consists of an Elementarity schema that asserts j is an elementary embedding, a Critical Point axiom that asserts existence of a least ordinal moved, and a schema Separation j that asserts Separation holds for all instances of j -formulas. The theory ZFC + WA has been proposed in the author’s earlier papers as a natural axiomatic extension of ZFC to account for most of the known large cardinals. In this paper we offer evidence for the naturalness of this theory by showing that it is, like ZFC itself, indestructible by set forcing. We show first that if κ is the critical point of the embedding, then ZFC+WA is preserved by any notion of forcing that belongs to V κ . This step is nontrivial because to prove Separation j holds in the forcing extension after lifting the embedding, it is necessary to incorporate j into the definition of the forcing relation. Then for arbitrary notions of forcing, we introduce a different technique of lifting that lifts one of the original embedding’s applicative iterates.
{"title":"Indestructibility of Wholeness","authors":"P. Corazza","doi":"10.4064/fm604-3-2020","DOIUrl":"https://doi.org/10.4064/fm604-3-2020","url":null,"abstract":". The Wholeness Axiom (WA) is an axiom schema that asserts the existence of a nontrivial elementary embedding from V to itself. The schema is formulated in the language {∈ , j } , where j is a unary function symbol intended to stand for the embedding. WA consists of an Elementarity schema that asserts j is an elementary embedding, a Critical Point axiom that asserts existence of a least ordinal moved, and a schema Separation j that asserts Separation holds for all instances of j -formulas. The theory ZFC + WA has been proposed in the author’s earlier papers as a natural axiomatic extension of ZFC to account for most of the known large cardinals. In this paper we offer evidence for the naturalness of this theory by showing that it is, like ZFC itself, indestructible by set forcing. We show first that if κ is the critical point of the embedding, then ZFC+WA is preserved by any notion of forcing that belongs to V κ . This step is nontrivial because to prove Separation j holds in the forcing extension after lifting the embedding, it is necessary to incorporate j into the definition of the forcing relation. Then for arbitrary notions of forcing, we introduce a different technique of lifting that lifts one of the original embedding’s applicative iterates.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70353211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. This paper presents an example, as promised by the title, of a degree-zero, monotone, surjective map of the standard Pontryagin surface to itself. This exposes the need for some hypothesis concerning degree in results about when monotone self-maps of the Pontryagin surface can be approximated by homeomorphisms.
{"title":"A degree-zero, monotone, surjective self-map of the Pontryagin surface","authors":"R. Daverman, T. L. Thickstun","doi":"10.4064/FM766-1-2021","DOIUrl":"https://doi.org/10.4064/FM766-1-2021","url":null,"abstract":". This paper presents an example, as promised by the title, of a degree-zero, monotone, surjective map of the standard Pontryagin surface to itself. This exposes the need for some hypothesis concerning degree in results about when monotone self-maps of the Pontryagin surface can be approximated by homeomorphisms.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70382172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An ultrapower analogue of the Kronecker function ring","authors":"K. Loper, D. Spirito","doi":"10.4064/fm833-4-2020","DOIUrl":"https://doi.org/10.4064/fm833-4-2020","url":null,"abstract":"We introduce an analogue of the Kronecker function ring construction in the ultrapower setting, and study when it gives a Bézout domain.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A new take on random interval homeomorphisms","authors":"J. Morawiec, T. Zürcher","doi":"10.4064/fm904-10-2021","DOIUrl":"https://doi.org/10.4064/fm904-10-2021","url":null,"abstract":"","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70402301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the interrelation of a theorem of Juhász and certain weak axioms of choice","authors":"E. Tachtsis","doi":"10.4064/fm37-2-2021","DOIUrl":"https://doi.org/10.4064/fm37-2-2021","url":null,"abstract":"","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70308723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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