Annals of Mathematical Logic最新文献_第5页

The model theory of ‘R-formal’ fields “r -形式”场的模型理论

Annals of Mathematical Logic Pub Date : 1980-12-01 DOI: 10.1016/0003-4843(80)90012-1

Bill Jacob

{"title":"The model theory of ‘R-formal’ fields","authors":"Bill Jacob","doi":"10.1016/0003-4843(80)90012-1","DOIUrl":"https://doi.org/10.1016/0003-4843(80)90012-1","url":null,"abstract":"<div>Let K be a field, and let W(K) denote its Witt ring of Quadratic Forms. It is well-known in the theory of Quadratic Forms that the orders of K correpond in a one to one way with all ring surjections <math><mtext>W(K) → </mtext><mtext>Z</mtext></math>. In particular, a field L is formally real over an ordered field K if and only if there is a homomorphism <math><mtext>ϕ</mtext><msub><mi></mi><mn>1</mn></msub><mtext>: W(L)→</mtext><mtext>Z</mtext></math> which extends the given ‘signature’ <math><mtext>ϕ</mtext><msub><mi></mi><mn>K</mn></msub><mtext>: W(K)→</mtext><mtext>Z</mtext></math>. (E.g. <math><mtext>ϕ</mtext><msub><mi></mi><mn>K</mn></msub><mtext> = ϕ</mtext><msub><mi></mi><mn>1</mn></msub><mtext>, i</mtext><msub><mi></mi><mn>∗</mn></msub><mtext>, </mtext><mtext>where</mtext><mtext> i</mtext><msub><mi></mi><mn>∗</mn></msub><mtext>: W(K)1 → W(L)</mtext></math> is the functinal map.)Using the above, one may discuss the usual theory of formally real and real closed fields in terms of Witt rings, Knebusch in [6] has, in the above setting, given a remarkable new proof of the uniqueness of real closures. One might ask what happens when the <math><mtext>Z</mtext></math> above is replaced by some other ring R? That is the subject of this present note. In particular, we shall prove some algebraic and model theoretic analogues of well-known results for real closed fields, where the above <math><mtext>Z</mtext></math> is replaced by some finitely generated reduced Witt ring.</div>","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"19 3","pages":"Pages 263-282"},"PeriodicalIF":0.0,"publicationDate":"1980-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(80)90012-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91721875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

Thin collections of sets of projective ordinals and analogs of L L的射影序数和类似物集合的细集合

Annals of Mathematical Logic Pub Date : 1980-12-01 DOI: 10.1016/0003-4843(80)90010-8

Howard Becker

引用次数: 26

Simple unstable theories 简单不稳定理论

Annals of Mathematical Logic Pub Date : 1980-12-01 DOI: 10.1016/0003-4843(80)90009-1

S. Shelah

引用次数: 201

The model theory of ‘R-formal’ fields “r -形式”场的模型理论

Annals of Mathematical Logic Pub Date : 1980-12-01 DOI: 10.1016/0003-4843(80)90012-1

B. Jacob

引用次数: 1

On the elementary theory of quadruples of vector spaces 关于向量空间的四重组的基本理论

Annals of Mathematical Logic Pub Date : 1980-12-01 DOI: 10.1016/0003-4843(80)90011-X

Walter Baur

引用次数: 19

Countable models of ω1-categorical theories in admissible languages 可容许语言中ω - 1-直言理论的可数模型

Annals of Mathematical Logic Pub Date : 1980-11-01 DOI: 10.1016/0003-4843(80)90023-6

Henry A. Kierstead

引用次数: 6

Perfect-set forcing for uncountable cardinals 不可数基数的完美设置强制

Annals of Mathematical Logic Pub Date : 1980-11-01 DOI: 10.1016/0003-4843(80)90021-2

Akihiro Kanamori

引用次数: 18

Interpretations of Heyting's arithmetic—An analysis by means of a language with set symbols 何亭算术的解读——用集合符号的语言分析

Annals of Mathematical Logic Pub Date : 1980-11-01 DOI: 10.1016/0003-4843(80)90018-2

Martin Stein

{"title":"Interpretations of Heyting's arithmetic—An analysis by means of a language with set symbols","authors":"Martin Stein","doi":"10.1016/0003-4843(80)90018-2","DOIUrl":"10.1016/0003-4843(80)90018-2","url":null,"abstract":"<div>Well-known interpretations of Heyting's arithmetic of all finite types are the Diller-Nahm λ-interpretation [1] and Kreisel's modified realizability, subsequently called mr-interpretation [4]. For both interpretations one can define hybrids λ q resp. mq.In Section 4 a chain of interpretations—called M-interpretations—is defined (it was introduced in [6], filling the “gap” between λ-interpretation and mr-interpretation.In this paper it is shwon that it is possible to prove in one stroke the soundness resp. characterization theorems for all interpretations of HAω 〈〉 (Heyting's arithmetic of all finite types with functionals for coding finite sequences). This is done by means of interpretations of systems which contain set-symbols. For these so called M-interpretations, soundness-resp. characterization theorems can be proved simultaneously (Theorem 2.51. Special translations of set symbols and of the formula (λωϵW)A — this means, special decisions about the size of the set W; see Sections 3 and 4 — yield the corresponding results for all interpretations of HAω〈〉 mentioned.The terminology of set theoretical language — we consider an extension of HAω〈〉 by an extensively weak fragment only, which leads to a conservative extension of HAω〈〉 — is of good use for studying realizing terms of different interpretations: if HAω<>⊢A, AM∃υ ∀w AM, and ⊢AM[tM, w] by soundness theorem for M-interpretations, there exists a simple operation which maps <math><mtext>v</mtext><mtext>̄</mtext><mtext> </mtext><mtext>to</mtext><mtext> </mtext><mtext>t</mtext><mtext>̄</mtext><msub><mi></mi><mn><mtext>mr</mtext></mn></msub></math>, the realizing term for modified realizability. For interpretations of Heyting's arithmetic — λ-interpretation. M-interpretations and mr-interpretation — this leads to the following stability result for existence theorems: if <math><mtext>∃λ A </mtext><mtext>and</mtext><mtext> t</mtext><msub><mi></mi><mn>^</mn></msub><mtext> </mtext><mtext>resp.</mtext><mtext> t</mtext><msub><mi></mi><mn><mtext>M</mtext><mtext>M</mtext></mn></msub></math> is the term computed by λ-interpretation. resp. M-interpretation, with <math><mtext>∃A[t</mtext><msub><mi></mi><mn><mtext>M</mtext></mn></msub><mtext>]</mtext></math>, then — using extensional equality and ω-rule for equations — we can prove that <math><mtext>t</mtext><msub><mi></mi><mn>λ</mn></msub><mtext> = t</mtext><msub><mi></mi><mn><mtext>M</mtext></mn></msub><mtext> = t</mtext><msub><mi></mi><mn><mtext>mr</mtext></mn></msub></math> (Section 5).</div>","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"19 1","pages":"Pages 1-31"},"PeriodicalIF":0.0,"publicationDate":"1980-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(80)90018-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87664539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 11

Derived rules related to a constructive theory of metric spaces in intuitionistic higher order arithmetic without countable choice 导出无可数选择的直觉高阶算术度量空间构造理论的规则

Annals of Mathematical Logic Pub Date : 1980-11-01 DOI: 10.1016/0003-4843(80)90019-4

Susumu Hayashi

引用次数: 5

Concerning the consistency of the Souslin hypothesis with the continuum hypothesis 关于苏斯林假设与连续统假设的一致性

Annals of Mathematical Logic Pub Date : 1980-11-01 DOI: 10.1016/0003-4843(80)90022-4

Keith J. Devlin

引用次数: 4