Trends in Set Theory最新文献

{"title":"The Ultrapower Axiom","authors":"G. Goldberg","doi":"10.1090/conm/752/15131","DOIUrl":"https://doi.org/10.1090/conm/752/15131","url":null,"abstract":"The inner model problem for supercompact cardinals, one of the central open problems in modern set theory, asks whether there is a canonical model of set theory with a supercompact cardinal. The problem is closely related to the more precise question of the equiconsistency of strongly compact cardinals and supercompact cardinals. This dissertation approaches these two problems abstractly by introducing a principle called the Ultrapower Axiom which is expected to hold in all known canonical models of set theory. By investigating the consequences of the Ultrapower Axiom under the hypothesis that there is a supercompact cardinal, we provide evidence that the inner model problem can be solved. Moreover, we establish that under the Ultrapower Axiom, strong compactness and supercompactness are essentially equivalent.","PeriodicalId":337244,"journal":{"name":"Trends in Set Theory","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114726284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An introduction to combinatorics of\u0000 determinacy","authors":"William Chan","doi":"10.1090/conm/752/15128","DOIUrl":"https://doi.org/10.1090/conm/752/15128","url":null,"abstract":"This article is an introduction to combinatorics under the axiom of determinacy with a focus on partition properties and infinity Borel codes.","PeriodicalId":337244,"journal":{"name":"Trends in Set Theory","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131227597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Distance from marker sequences in locally\u0000 finite Borel graphs","authors":"Clinton T. Conley, Andrew S. Marks","doi":"10.1090/conm/752/15130","DOIUrl":"https://doi.org/10.1090/conm/752/15130","url":null,"abstract":"We show that a locally finite Borel graph is nonsmooth if and only if it admits marker sequences which are \"far\" from every point. Our proof uses the Galvin-Prikry theorem and the Glimm-Effros dichotomy.","PeriodicalId":337244,"journal":{"name":"Trends in Set Theory","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124385392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A brief introduction to amenable equivalence\u0000 relations","authors":"J. Moore","doi":"10.1090/conm/752/15134","DOIUrl":"https://doi.org/10.1090/conm/752/15134","url":null,"abstract":"The notion of an amenable equivalence relation was introduced by Zimmer in the course of his analysis of orbit equivalence relations in ergodic theory (see [12]). More recently it played an important role in Monod’s striking family of examples of nonamenable groups which do not contain nonabelian free subgroups. If A is a subring of R, define H(A) to be the group of all piecewise PSL2(A) homeomorphisms of the real projective line which fix the point at infinity.","PeriodicalId":337244,"journal":{"name":"Trends in Set Theory","volume":"37 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120969510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A descriptive construction of trees and\u0000 Stallings’ theorem","authors":"A. Tserunyan","doi":"10.1090/conm/752/15137","DOIUrl":"https://doi.org/10.1090/conm/752/15137","url":null,"abstract":"We give a descriptive construction of trees for multi-ended graphs, which yields yet another proof of Stallings' theorem on ends of groups. Even though our proof is, in principle, not very different from already existing proofs and it draws ideas from a paper of Kr\"{o}n, it is written in a way that easily adapts to the setting of countable Borel equivalence relations, leading to a free decomposition result and a sufficient condition for treeability.","PeriodicalId":337244,"journal":{"name":"Trends in Set Theory","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132476077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An example of a new simple theory","authors":"M. Malliaris, S. Shelah","doi":"10.1090/conm/752/15133","DOIUrl":"https://doi.org/10.1090/conm/752/15133","url":null,"abstract":"We construct a countable simple theory which, in Keisler's order, is strictly above the random graph (but \"barely so\") and also in some sense orthogonal to the building blocks of the recently discovered infinite descending chain. As a result we prove in ZFC that there are incomparable classes in Keisler's order.","PeriodicalId":337244,"journal":{"name":"Trends in Set Theory","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124889310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The open dihypergraph dichotomy and the\u0000 second level of the Borel hierarchy","authors":"Raphaël Carroy, B. D. Miller, D. Soukup","doi":"10.1090/conm/752/15127","DOIUrl":"https://doi.org/10.1090/conm/752/15127","url":null,"abstract":"We show that several dichotomy theorems concerning the second level of the Borel hierarchy are special cases of the $aleph_0$-dimensional generalization of the open graph dichotomy, which itself follows from the usual proof(s) of the perfect set theorem. Under the axiom of determinacy, we obtain the generalizations of these results from analytic metric spaces to separable metric spaces. We also consider connections between cardinal invariants and the chromatic numbers of the corresponding dihypergraphs.","PeriodicalId":337244,"journal":{"name":"Trends in Set Theory","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130602081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Singularizing cardinals","authors":"Dima Sinapova","doi":"10.1090/conm/752/15135","DOIUrl":"https://doi.org/10.1090/conm/752/15135","url":null,"abstract":"","PeriodicalId":337244,"journal":{"name":"Trends in Set Theory","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125012636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A brief account of recent developments in\u0000 inner model theory","authors":"Nam Trang","doi":"10.1090/conm/752/15136","DOIUrl":"https://doi.org/10.1090/conm/752/15136","url":null,"abstract":"The goal of this survey paper is to give an overview of recent developments in inner model theory. We discuss several most important questions in the field and relevant work (partially) answering these questions roughly from the early 2000’s to present. Most of the major questions are still open. These questions drive future research directions in inner model theory. We mention some of the most prominent projects pertaining to these.1 Inner model theory starts with Gödel’s work on the constructible universe L in 1938. L is the first example of a “canonical inner model”. There is no official definition of “canonical inner models”, but in the case of L, it is canonical because it is the minimal transitive proper class model of ZFC and it satisfies various benchmark properties that one would want canonical inner models to satisfy, such as the Generalized Continuum Hypothesis (GCH), 3 and 2 (the latter two are due to Jensen in his seminal work [8]). L, however, does not admit very large large cardinals. For instance, Scott, [31], has shown that L cannot contain a measurable cardinal. A main theme in inner model theory after Gödel’s work and Scott’s result is the search for canonical inner models that can accommodate large cardinals beyond those accommodated by L. This is sometimes known as the Gödel program. An immediate target is a measurable cardinal. Models of the form L[U ] that satisfy “U is a κ-complete, normal, nonprincipal measure on a cardinal κ” have been extensively studied. Most notably, it can be shown that L[U ] satisfies the GCH (this is due to Silver [32]), 3,2. Kunen, in [11], develops the influential theory of M -ultrapowers (based on previous work by H. Gaifman) and uses it to prove fundamental results on the canonicity of L[U ]. In particular, he shows that L[U ] is unique internally (namely U is the unique normal measure on κ in L[U ]) and for i ∈ {0, 1}, if Ui is a filter on κ such that L[Ui] “Ui is a normal measure on κ”, then L[U0] = L[U1] and U0 ∩ L[U0] = U1 ∩ L[U1]. At the heart of Kunen’s analysis is the comparability of two models of the form L[U ], i.e. for i ∈ {0, 1}, if Mi = L[Ui] “Ui is a normal measure on κi”, then either M0 is an iterated ultrapower of M1 or M1 is an iterated ultrapower of M0 (via the measure Ui and its images). Mitchell, in [14] and [15], generalizes Kunen’s analysis to models of the form L[~ U ] where ~ U is a sequence of measures. Mitchell’s definition of an extender (see Section 4 for a definition) then plays a key role in subsequent searches for canonical inner models of larger large cardinals such This paper is not meant to be a comprehensive encyclopedia of the subject. We apologize for any omissions of recent works that are not mentioned here.","PeriodicalId":337244,"journal":{"name":"Trends in Set Theory","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126266143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}