{"title":"集理论","authors":"Jason W. Solomon","doi":"10.4324/9781315167749-28","DOIUrl":null,"url":null,"abstract":"0009 Most mathematicians think of set theory as providing the basic foundations for mathematics. So how does this really work? For example, how do we translate the sentence “X is a scheme” into set theory? Well, we just unravel the definitions: A scheme is a locally ringed space such that every point has an open neighbourhood which is an affine scheme. A locally ringed space is a ringed space such that every stalk of the structure sheaf is a local ring. A ringed space is a pair (X,OX) consisting of a topological space X and a sheaf of rings OX on it. A topological space is a pair (X, τ) consisting of a set X and a set of subsets τ ⊂ P(X) satisfying the axioms of a topology. And so on and so forth.","PeriodicalId":207895,"journal":{"name":"Music Theory Essentials","volume":"30 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Set Theory\",\"authors\":\"Jason W. Solomon\",\"doi\":\"10.4324/9781315167749-28\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"0009 Most mathematicians think of set theory as providing the basic foundations for mathematics. So how does this really work? For example, how do we translate the sentence “X is a scheme” into set theory? Well, we just unravel the definitions: A scheme is a locally ringed space such that every point has an open neighbourhood which is an affine scheme. A locally ringed space is a ringed space such that every stalk of the structure sheaf is a local ring. A ringed space is a pair (X,OX) consisting of a topological space X and a sheaf of rings OX on it. A topological space is a pair (X, τ) consisting of a set X and a set of subsets τ ⊂ P(X) satisfying the axioms of a topology. And so on and so forth.\",\"PeriodicalId\":207895,\"journal\":{\"name\":\"Music Theory Essentials\",\"volume\":\"30 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-03-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Music Theory Essentials\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4324/9781315167749-28\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Music Theory Essentials","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4324/9781315167749-28","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
0009 Most mathematicians think of set theory as providing the basic foundations for mathematics. So how does this really work? For example, how do we translate the sentence “X is a scheme” into set theory? Well, we just unravel the definitions: A scheme is a locally ringed space such that every point has an open neighbourhood which is an affine scheme. A locally ringed space is a ringed space such that every stalk of the structure sheaf is a local ring. A ringed space is a pair (X,OX) consisting of a topological space X and a sheaf of rings OX on it. A topological space is a pair (X, τ) consisting of a set X and a set of subsets τ ⊂ P(X) satisfying the axioms of a topology. And so on and so forth.
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