{"title":"On the length of Borel hierarchies","authors":"Arnorld W. Miller","doi":"10.1016/0003-4843(79)90003-2","DOIUrl":"10.1016/0003-4843(79)90003-2","url":null,"abstract":"","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"16 3","pages":"Pages 233-267"},"PeriodicalIF":0.0,"publicationDate":"1979-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(79)90003-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85609417","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}
{"title":"Degrees of functionals","authors":"Dag Normann","doi":"10.1016/0003-4843(79)90004-4","DOIUrl":"10.1016/0003-4843(79)90004-4","url":null,"abstract":"<div><p>In this paper we will discuss some problems of degree-theoretic nature in connection with recursion in normal objects of higher types.</p><p>Harrington [2] and Loewenthal [6] have proved some results concerning Post's problem and the Minimal Pair Problem, using recursion modulo subindividuals. Our degrees will be those obtained from Kleene-recursion modulo individuals. To solve our problems we then have to put some extra strength to ZFC. We will first assume <em>V</em> = <em>L</em>, and then we restrict ourselves to the situation of a recursive well-ordering and Martin's axiom.</p><p>We assume familiarity with recursion theory in higher types as presented in Kleene [3]. Further backround is found in Harrington [2], Moldestad [9] and Normann [11]. We will survey the parts of these papers that we need.</p><p>In Section 1 we give the general background for the arguments used later. In Section 2 we prove some lemmas assuming <em>V</em> = <em>L</em>. In section 3, assuming <em>V</em> = <em>L</em> we solve Post's problem and another problem using the finite injury method. We will thereby describe some of the methods needed for the more complex priority argument of Section 4 where we give a solution of the minimal pair problem for extended r.e. degress of functionals.</p><p>In Section 5 we will see that if Martin's Axiom holds and we have a minimal well-ordering of tp (1) recursive in <sup>3</sup><em>E</em>, we may use the same sort of arguments as in parts 3 and 4.</p></div>","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"16 3","pages":"Pages 269-304"},"PeriodicalIF":0.0,"publicationDate":"1979-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(79)90004-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84820328","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}
{"title":"Substructure lattices of models of arithmetic","authors":"George Mills","doi":"10.1016/0003-4843(79)90007-X","DOIUrl":"10.1016/0003-4843(79)90007-X","url":null,"abstract":"<div><p>We completely characterize those distributive lattices which can be obtained as elementary substructure lattices of models of Peano arithmetic. Stated concisely: every plausible distributive lattice occurs abundantly. Our proof employs the notion of a strongly definable type in many variables. With slight modifications the method also yields a characterization of those distributive lattices which can be obtained uniformly by Gaifman's methods of definable and end extensional 1-types. As a special case this gives another proof of two conjectures involving finite distributive lattices and models of arithmetic posed by Gaifman and initially proved by Schmerl. We also show that every minimal type (in the sense of Gaifman) satisfies a strong partition property which we will call being “uniformly Ramsey”.</p></div>","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"16 2","pages":"Pages 145-180"},"PeriodicalIF":0.0,"publicationDate":"1979-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(79)90007-X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86601460","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}
{"title":"Some applications of Kripke models to formal systems of intuitionistic analysis","authors":"Scott Weinstein","doi":"10.1016/0003-4843(79)90014-7","DOIUrl":"10.1016/0003-4843(79)90014-7","url":null,"abstract":"","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"16 1","pages":"Pages 1-32"},"PeriodicalIF":0.0,"publicationDate":"1979-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(79)90014-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87118284","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}
{"title":"Regularity properties of ideals and ultrafilters","authors":"Alan D. Taylor","doi":"10.1016/0003-4843(79)90015-9","DOIUrl":"10.1016/0003-4843(79)90015-9","url":null,"abstract":"<div><p>For an arbitrary ideal <em>I</em> on the regular cardinal κ we consider the problem of refining a given collection <span><math><mtext>{</mtext><mtext>A</mtext><msub><mi></mi><mn>α</mn></msub><mtext>:α < κ}⊆</mtext><mtext>P</mtext><mtext>(κ)−I</mtext></math></span> by another collection <span><math><mtext>{</mtext><mtext>B</mtext><msub><mi></mi><mn>α</mn></msub><mtext>:α<κ}⊆</mtext><mtext>P</mtext><mtext>(k)−I</mtext></math></span> so that the sets in the latter collection are as nearly pairwise disjoint as possible. In this context we discuss regularity of ultrafilters, saturation of ideals and some problems of Fodor and Ulam.</p></div>","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"16 1","pages":"Pages 33-55"},"PeriodicalIF":0.0,"publicationDate":"1979-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(79)90015-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83253463","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}
{"title":"Generalized erdoös cardinals and O4","authors":"James E. Baumgartner , Fred Galvin","doi":"10.1016/0003-4843(78)90012-8","DOIUrl":"10.1016/0003-4843(78)90012-8","url":null,"abstract":"","PeriodicalId":100093,"journal":{"name":"Annals of Mathematical Logic","volume":"15 3","pages":"Pages 289-313"},"PeriodicalIF":0.0,"publicationDate":"1978-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0003-4843(78)90012-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76109353","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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