Annals of Pure and Applied Logic最新文献

{"title":"Generalized independence","authors":"Fernando Hernández-Hernández ,&nbsp;Carlos López-Callejas","doi":"10.1016/j.apal.2024.103440","DOIUrl":"10.1016/j.apal.2024.103440","url":null,"abstract":"<div><p>We explore different generalizations of the classical concept of independent families on <em>ω</em> following the study initiated by Kunen, Fischer, Eskew and Montoya. We show that under <span><math><msubsup><mrow><mo>(</mo><mi>D</mi><mi>ℓ</mi><mo>)</mo></mrow><mrow><mi>κ</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> we can get strongly <em>κ</em>-independent families of size <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>κ</mi></mrow></msup></math></span> and present an equivalence of <span><math><mi>GCH</mi></math></span> in terms of strongly independent families. We merge the two natural ways of generalizing independent families through a filter or an ideal and we focus on the <span><math><mi>C</mi></math></span>-independent families, where <span><math><mi>C</mi></math></span> is the club filter. Also we show a relationship between the existence of <span><math><mi>J</mi></math></span>-independent families and the saturation of the ideal <span><math><mi>J</mi></math></span>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 7","pages":"Article 103440"},"PeriodicalIF":0.8,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140274442","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Big Ramsey degrees in ultraproducts of finite structures","authors":"Dana Bartošová ,&nbsp;Mirna Džamonja ,&nbsp;Rehana Patel ,&nbsp;Lynn Scow","doi":"10.1016/j.apal.2024.103439","DOIUrl":"10.1016/j.apal.2024.103439","url":null,"abstract":"<div><p>We develop a transfer principle of structural Ramsey theory from finite structures to ultraproducts. We show that under certain mild conditions, when a class of finite structures has finite small Ramsey degrees, under the (Generalized) Continuum Hypothesis the ultraproduct has finite big Ramsey degrees for internal colorings. The necessity of restricting to internal colorings is demonstrated by the example of the ultraproduct of finite linear orders. Under CH, this ultraproduct <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> has, as a spine, <span><math><msub><mrow><mi>η</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, an uncountable analogue of the order type of rationals <em>η</em>. Finite big Ramsey degrees for <em>η</em> were exactly calculated by Devlin in <span>[5]</span>. It is immediate from <span>[39]</span> that <span><math><msub><mrow><mi>η</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> fails to have finite big Ramsey degrees. Moreover, we extend Devlin's coloring to <span><math><msub><mrow><mi>η</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> to show that it witnesses big Ramsey degrees of finite tuples in <em>η</em> on every copy of <em>η</em> in <span><math><msub><mrow><mi>η</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, and consequently in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. This work gives additional confirmation that ultraproducts are a suitable environment for studying Ramsey properties of finite and infinite structures.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 7","pages":"Article 103439"},"PeriodicalIF":0.8,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000368/pdfft?md5=96a2fc37ad227ed1f90534ab7367f0e2&pid=1-s2.0-S0168007224000368-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140181925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Higher reciprocity law and an analogue of the Grunwald–Wang theorem for the ring of polynomials over an ultra-finite field","authors":"Dong Quan Ngoc Nguyen","doi":"10.1016/j.apal.2024.103438","DOIUrl":"10.1016/j.apal.2024.103438","url":null,"abstract":"<div><p>In this paper, we establish an explicit higher reciprocity law for the polynomial ring over a nonprincipal ultraproduct of finite fields. Such an ultraproduct can be taken over the same finite field, which allows to recover the classical higher reciprocity law for the polynomial ring <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>t</mi><mo>]</mo></math></span> over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> that is due to Dedekind, Kühne, Artin, and Schmidt. On the other hand, when the ultraproduct is taken over finite fields of unbounded cardinalities, we obtain an explicit higher reciprocity law for the polynomial ring over an infinite field in both characteristics 0 and <span><math><mi>p</mi><mo>&gt;</mo><mn>0</mn></math></span> for some prime <em>p</em>. We then use the higher reciprocity law to prove an analogue of the Grunwald–Wang theorem for such a polynomial ring in both characteristics 0 and <span><math><mi>p</mi><mo>&gt;</mo><mn>0</mn></math></span> for some prime <em>p</em>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 6","pages":"Article 103438"},"PeriodicalIF":0.8,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140181908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approachable free subsets and fine structure derived scales","authors":"Dominik Adolf,&nbsp;Omer Ben-Neria","doi":"10.1016/j.apal.2024.103428","DOIUrl":"10.1016/j.apal.2024.103428","url":null,"abstract":"<div><p>Shelah showed that the existence of free subsets over internally approachable subalgebras follows from the failure of the PCF conjecture on intervals of regular cardinals. We show that a stronger property called the Approachable Bounded Subset Property can be forced from the assumption of a cardinal <em>λ</em> for which the set of Mitchell orders <span><math><mo>{</mo><mi>o</mi><mo>(</mo><mi>μ</mi><mo>)</mo><mo>|</mo><mi>μ</mi><mo>&lt;</mo><mi>λ</mi><mo>}</mo></math></span> is unbounded in <em>λ</em>. Furthermore, we study the related notion of continuous tree-like scales, and show that such scales must exist on all products in canonical inner models. We use this result, together with a covering-type argument, to show that the large cardinal hypothesis from the forcing part is optimal.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 7","pages":"Article 103428"},"PeriodicalIF":0.8,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140181769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"ZF and its interpretations","authors":"S. Jockwich Martinez ,&nbsp;S. Tarafder ,&nbsp;G. Venturi","doi":"10.1016/j.apal.2024.103427","DOIUrl":"10.1016/j.apal.2024.103427","url":null,"abstract":"<div><p>In this paper, we unify the study of classical and non-classical algebra-valued models of set theory, by studying variations of the interpretation functions for = and ∈. Although, these variations coincide with the standard interpretation in Boolean-valued constructions, nonetheless they extend the scope of validity of <span><math><mi>ZF</mi></math></span> to new algebra-valued models. This paper presents, for the first time, non-trivial paraconsistent models of full <span><math><mi>ZF</mi></math></span>. Moreover, due to the validity of Leibniz's law in these structures, we will show how to construct proper models of set theory by quotienting these algebra-valued models with respect to equality, modulo the filter of the designated truth-values.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 6","pages":"Article 103427"},"PeriodicalIF":0.8,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000241/pdfft?md5=9d7bf0eef51dc942a71051fb8dfcc3b5&pid=1-s2.0-S0168007224000241-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A good lightface Δn1 well-ordering of the reals does not imply the existence of boldface Δn−11 well-orderings","authors":"Vladimir Kanovei,&nbsp;Vassily Lyubetsky","doi":"10.1016/j.apal.2024.103426","DOIUrl":"10.1016/j.apal.2024.103426","url":null,"abstract":"<div><p>We make use of a finite support product of the Jensen-type forcing notions to define a model of the set theory <span><math><mtext>ZFC</mtext></math></span> in which, for a given <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, there exists a good lightface <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> well-ordering of the reals but there are no any (not necessarily good) well-orderings in the boldface class <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 6","pages":"Article 103426"},"PeriodicalIF":0.8,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nonstandard proof methods in toposes","authors":"José Siqueira","doi":"10.1016/j.apal.2024.103424","DOIUrl":"10.1016/j.apal.2024.103424","url":null,"abstract":"<div><p>We determine sufficient structure for an elementary topos to emulate Nelson's Internal Set Theory in its internal language, and show that any topos satisfying the internal axiom of choice occurs as a universe of standard objects and maps. This development allows one to employ the proof methods of nonstandard analysis (transfer, standardisation, and idealisation) in new environments such as toposes of <em>G</em>-sets and Boolean étendues.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 5","pages":"Article 103424"},"PeriodicalIF":0.8,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139946240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Arboreal categories and equi-resource homomorphism preservation theorems","authors":"Samson Abramsky,&nbsp;Luca Reggio","doi":"10.1016/j.apal.2024.103423","DOIUrl":"10.1016/j.apal.2024.103423","url":null,"abstract":"<div><p>The classical homomorphism preservation theorem, due to Łoś, Lyndon and Tarski, states that a first-order sentence <em>φ</em> is preserved under homomorphisms between structures if, and only if, it is equivalent to an existential positive sentence <em>ψ</em>. Given a notion of (syntactic) complexity of sentences, an “equi-resource” homomorphism preservation theorem improves on the classical result by ensuring that <em>ψ</em> can be chosen so that its complexity does not exceed that of <em>φ</em>.</p><p>We describe an axiomatic approach to equi-resource homomorphism preservation theorems based on the notion of arboreal category. This framework is then employed to establish novel homomorphism preservation results, and improve on known ones, for various logic fragments, including first-order, guarded and modal logics.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 6","pages":"Article 103423"},"PeriodicalIF":0.8,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000204/pdfft?md5=483bf3d114b061fe423ab82a314823cd&pid=1-s2.0-S0168007224000204-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139925039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A new model construction by making a detour via intuitionistic theories IV: A closer connection between KPω and BI","authors":"Kentaro Sato","doi":"10.1016/j.apal.2024.103422","DOIUrl":"10.1016/j.apal.2024.103422","url":null,"abstract":"<div><p>By combining tree representation of sets with the method introduced in the previous three papers I–III <span>[39]</span>, <span>[35]</span>, <span>[37]</span> in the series, we give a new <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-preserving interpretation of <span><math><mrow><mi>KP</mi></mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>+</mo><mo>(</mo><msub><mrow><mi>Π</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub><mtext>-</mtext><mrow><mi>Found</mi></mrow><mo>)</mo><mo>+</mo><mi>θ</mi></math></span> (Kripke–Platek set theory with the foundation schema restricted to <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span>, and augmented by <em>θ</em>) in <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mtext>-</mtext><msub><mrow><mi>AC</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>+</mo><mo>(</mo><msubsup><mrow><mi>Π</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mtext>-</mtext><mrow><mi>TI</mi></mrow><mo>)</mo><mo>+</mo><mi>θ</mi></math></span> for any <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> sentence <em>θ</em>, where the language of second order arithmetic is considered as a sublanguage of that of set theory via the standard interpretation. Thus the addition of any <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> theorem of <span><math><mrow><mi>BI</mi></mrow><mo>≡</mo><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mtext>-</mtext><msub><mrow><mi>AC</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>+</mo><mo>(</mo><msubsup><mrow><mi>Π</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mn>1</mn></mrow></msubsup><mtext>-</mtext><mrow><mi>TI</mi></mrow><mo>)</mo></math></span> does not increase the consistency strength of <strong>KP</strong><em>ω</em>. Among such <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> theorems are several fixed point principles for positive arithmetical operators and <em>ω</em>-model reflection (the cofinal existence of coded <em>ω</em>-models) for theorems of <strong>BI</strong>. The reader's familiarity to the previous works I–III in the series might help, but is not necessary.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 7","pages":"Article 103422"},"PeriodicalIF":0.8,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139811673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Strong ergodicity phenomena for Bernoulli shifts of bounded algebraic dimension","authors":"Aristotelis Panagiotopoulos ,&nbsp;Assaf Shani","doi":"10.1016/j.apal.2024.103412","DOIUrl":"10.1016/j.apal.2024.103412","url":null,"abstract":"<div><p>The algebraic dimension of a Polish permutation group <span><math><mi>Q</mi><mo>≤</mo><mrow><mi>Sym</mi></mrow><mo>(</mo><mi>N</mi><mo>)</mo></math></span> is the size of the largest <span><math><mi>A</mi><mo>⊆</mo><mi>N</mi></math></span> with the property that the orbit of every <span><math><mi>a</mi><mo>∈</mo><mi>A</mi></math></span> under the pointwise stabilizer of <span><math><mi>A</mi><mo>∖</mo><mo>{</mo><mi>a</mi><mo>}</mo></math></span> is infinite. We study the Bernoulli shift <span><math><mi>P</mi><mo>↷</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> for various Polish permutation groups <em>P</em> and we provide criteria under which the <em>P</em>-shift is generically ergodic relative to the injective part of the <em>Q</em>-shift, when <em>Q</em> has algebraic dimension ≤<em>n</em>. We use this to show that the sequence of pairwise ⁎-reduction-incomparable equivalence relations defined in <span>[18]</span> is a strictly increasing sequence in the Borel reduction hierarchy. We also use our main theorem to exhibit an equivalence relation of pinned cardinal <span><math><msubsup><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> which strongly resembles the equivalence relation of pinned cardinal <span><math><msubsup><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> from <span>[25]</span>, but which does not Borel reduce to the latter. It remains open whether they are actually incomparable under Borel reductions.</p><p>Our proofs rely on the study of symmetric models whose symmetries come from the group <em>Q</em>. We show that when <em>Q</em> is “locally finite”—e.g. when <span><math><mi>Q</mi><mo>=</mo><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>M</mi><mo>)</mo></math></span>, where <span><math><mi>M</mi></math></span> is the Fraïssé limit of a Fraïssé class satisfying the disjoint amalgamation property—the corresponding symmetric model admits a theory of supports which is analogous to that in the basic Cohen model.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 5","pages":"Article 103412"},"PeriodicalIF":0.8,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}