Pub Date : 1982-08-01DOI: 10.1016/0003-4843(82)90023-7
Lon Berk Radin
If κ is measurable, Prikry's forcing adds a sequence of ordinals of order type ω cofinal in κ. This destroys the regularity of κ but κ does remain uncountable. Magidor has a forcing notion generalizing Prikry's which adds a closed cofinal sequence of ordinals through a large cardinal. The cardinal remains uncountable but uts regularity is still destroyed. We obtain a forcing notion which adds a closed cofinal sequence of ordinals (and more complex objects) through a large cardinal κ, of order type κ, and keeps κ regular. In fact κ remains measurable after the forcing.
Our forcing shares certain properties with Prikry's forcing. Closed cofinal sebsequences of generic sequences are generic (under appropriate interpretations). Archetypical generic sequences can be generated by taking the critical points of iterated elementary embeddings.
{"title":"Adding closed cofinal sequences to large cardinals","authors":"Lon Berk Radin","doi":"10.1016/0003-4843(82)90023-7","DOIUrl":"10.1016/0003-4843(82)90023-7","url":null,"abstract":"<div><p>If κ is measurable, Prikry's forcing adds a sequence of ordinals of order type ω cofinal in κ. This destroys the regularity of κ but κ does remain uncountable. Magidor has a forcing notion generalizing Prikry's which adds a closed cofinal sequence of ordinals through a large cardinal. The cardinal remains uncountable but uts regularity is still destroyed. We obtain a forcing notion which adds a closed cofinal sequence of ordinals (and more complex objects) through a large cardinal κ, of order type κ, and keeps κ regular. In fact κ remains measurable after the forcing.</p><p>Our forcing shares certain properties with Prikry's forcing. Closed cofinal sebsequences of generic sequences are generic (under appropriate interpretations). Archetypical generic sequences can be generated by taking the critical points of iterated elementary embeddings.</p></div>","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"22 3","pages":"Pages 243-261"},"PeriodicalIF":0.0,"publicationDate":"1982-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(82)90023-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129995869","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}
Pub Date : 1982-08-01DOI: 10.1016/0003-4843(82)90022-5
Evangelos Kranakis
The present paper studies the relation between admissibility, reflection and partition properties. After introducing basic notions in Section e, Σn admissible ordinals are characterized using reflection properties (Section 2). Σn partition relations are introduced in Section 3. In Sections 3 and 4 connections are explored between partition properties, admissibility and projecta. Several more characterizations of admissibility are given in Section 5 (using Σn trees) and Section 6 (using Σn compactness). The ideas developed in Section 5 are used in Section 7 to study the partition relation .
{"title":"Reflection and partition properties of admissible ordinals","authors":"Evangelos Kranakis","doi":"10.1016/0003-4843(82)90022-5","DOIUrl":"10.1016/0003-4843(82)90022-5","url":null,"abstract":"<div><p>The present paper studies the relation between admissibility, reflection and partition properties. After introducing basic notions in Section e, <em>Σ</em><sub><em>n</em></sub> admissible ordinals are characterized using reflection properties (Section 2). <em>Σ</em><sub><em>n</em></sub> partition relations are introduced in Section 3. In Sections 3 and 4 connections are explored between partition properties, admissibility and projecta. Several more characterizations of admissibility are given in Section 5 (using <em>Σ</em><sub><em>n</em></sub> trees) and Section 6 (using <em>Σ</em><sub><em>n</em></sub> compactness). The ideas developed in Section 5 are used in Section 7 to study the partition relation <span><math><mtext>κ → </mtext><msup><mi></mi><mn><mtext>σ</mtext><msub><mi></mi><mn>n</mn></msub></mn></msup><mtext> (κ)</mtext><msup><mi></mi><mn>2</mn></msup></math></span>.</p></div>","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"22 3","pages":"Pages 213-242"},"PeriodicalIF":0.0,"publicationDate":"1982-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(82)90022-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121852227","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}
Pub Date : 1982-07-01DOI: 10.1016/0003-4843(82)90021-3
Daniele Mundici
Let Robinson's consistency theorem hold in logic L: then L will satisfy all the usual interpolation and definability properties, together with coutable compactness, provided L is reasonably small. The latter assumption can be weakened ro removed by using special set-theoretical assumptions. Thus, if Robinson's consistency theorem holds in L, then (i) L is countably compact if its Löwenheim number is < μ0 = the smallest uncountable measurable cardinal; (ii) if ω is the only measurable cardinal, L is countably compact, or the theories of L characterize every structure up to isomorphism. As a corollary, a partial answer is given to H. Friedman's third problem, by proving that no logic L strictly between L∞ω and L∞∞ satisfies interpolation (or Robinson's consistency), unless K-elementary equivalence coincides with isomorphism.
{"title":"Compactness, interpolation and Friedman's third problem","authors":"Daniele Mundici","doi":"10.1016/0003-4843(82)90021-3","DOIUrl":"https://doi.org/10.1016/0003-4843(82)90021-3","url":null,"abstract":"<div><p>Let Robinson's consistency theorem hold in logic <em>L</em>: then <em>L</em> will satisfy all the usual interpolation and definability properties, together with coutable compactness, provided <em>L</em> is reasonably small. The latter assumption can be weakened ro removed by using special set-theoretical assumptions. Thus, if Robinson's consistency theorem holds in <em>L</em>, then (i) <em>L</em> is countably compact if its Löwenheim number is < <em>μ</em><sub>0</sub> = the smallest uncountable measurable cardinal; (ii) if ω is the only measurable cardinal, <em>L</em> is countably compact, or the theories of <em>L</em> characterize every structure up to isomorphism. As a corollary, a partial answer is given to H. Friedman's third problem, by proving that no logic <em>L</em> strictly between <em>L</em><sub>∞<em>ω</em></sub> and <em>L</em><sub>∞∞</sub> satisfies interpolation (or Robinson's consistency), unless <em>K</em>-elementary equivalence coincides with isomorphism.</p></div>","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"22 2","pages":"Pages 197-211"},"PeriodicalIF":0.0,"publicationDate":"1982-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(82)90021-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90004582","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}
Pub Date : 1982-07-01DOI: 10.1016/0003-4843(82)90017-1
John R. Steel
{"title":"Determinacy in the Mitchell models","authors":"John R. Steel","doi":"10.1016/0003-4843(82)90017-1","DOIUrl":"https://doi.org/10.1016/0003-4843(82)90017-1","url":null,"abstract":"","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"22 2","pages":"Pages 109-125"},"PeriodicalIF":0.0,"publicationDate":"1982-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(82)90017-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91645908","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}
Pub Date : 1982-06-01DOI: 10.1016/0003-4843(82)90014-6
Sy D. Friedman
{"title":"Steel forcing and barwise compactness","authors":"Sy D. Friedman","doi":"10.1016/0003-4843(82)90014-6","DOIUrl":"10.1016/0003-4843(82)90014-6","url":null,"abstract":"","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"22 1","pages":"Pages 31-46"},"PeriodicalIF":0.0,"publicationDate":"1982-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(82)90014-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127359913","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}
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