Pub Date : 2025-03-19DOI: 10.1016/j.apal.2025.103583
Victoria Gitman , Jonathan Osinski
Galeotti, Khomskii and Väänänen recently introduced the notion of the upward Löwenheim-Skolem-Tarski number for a logic, strengthening the classical notion of a Hanf number. A cardinal κ is the upward Löwenheim-Skolem-Tarski number (ULST number) of a logic if it is the least cardinal with the property that whenever M is a model of size at least κ satisfying a sentence φ in , then there are arbitrarily large models satisfying φ and having M as a substructure. The substructure requirement is what differentiates the ULST number from the Hanf number and gives the notion large cardinal strength. While it is a theorem of ZFC that every logic has a Hanf number, Galeotti, Khomskii and Väänänen showed that the existence of the ULST number for second-order logic implies the existence of a partially extendible cardinal. We answer positively their conjecture that the ULST number for second-order logic is the least extendible cardinal.
We define the strong ULST number by strengthening the substructure requirement to elementary substructure. We investigate the ULST and strong ULST numbers for several classical strong logics: infinitary logics, the equicardinality logic, logic with the well-foundedness quantifier, second-order logic, and sort logics. We show that the ULST and the strong ULST numbers are characterized in some cases by classical large cardinals and in some cases by natural new large cardinal notions that they give rise to. We show that for some logics the notions of the ULST number, strong ULST number and least strong compactness cardinal coincide, while for others, it is consistent that they can be separated. Finally, we introduce a natural large cardinal notion characterizing strong compactness cardinals for the equicardinality logic.
{"title":"Upward Löwenheim-Skolem-Tarski numbers for abstract logics","authors":"Victoria Gitman , Jonathan Osinski","doi":"10.1016/j.apal.2025.103583","DOIUrl":"10.1016/j.apal.2025.103583","url":null,"abstract":"<div><div>Galeotti, Khomskii and Väänänen recently introduced the notion of the upward Löwenheim-Skolem-Tarski number for a logic, strengthening the classical notion of a Hanf number. A cardinal <em>κ</em> is the <em>upward Löwenheim-Skolem-Tarski number</em> (ULST <em>number</em>) of a logic <span><math><mi>L</mi></math></span> if it is the least cardinal with the property that whenever <em>M</em> is a model of size at least <em>κ</em> satisfying a sentence <em>φ</em> in <span><math><mi>L</mi></math></span>, then there are arbitrarily large models satisfying <em>φ</em> and having <em>M</em> as a substructure. The substructure requirement is what differentiates the ULST number from the Hanf number and gives the notion large cardinal strength. While it is a theorem of ZFC that every logic has a Hanf number, Galeotti, Khomskii and Väänänen showed that the existence of the ULST number for second-order logic implies the existence of a partially extendible cardinal. We answer positively their conjecture that the ULST number for second-order logic is the least extendible cardinal.</div><div>We define the <em>strong</em> ULST number by strengthening the substructure requirement to elementary substructure. We investigate the ULST and strong ULST numbers for several classical strong logics: infinitary logics, the equicardinality logic, logic with the well-foundedness quantifier, second-order logic, and sort logics. We show that the ULST and the strong ULST numbers are characterized in some cases by classical large cardinals and in some cases by natural new large cardinal notions that they give rise to. We show that for some logics the notions of the ULST number, strong ULST number and least strong compactness cardinal coincide, while for others, it is consistent that they can be separated. Finally, we introduce a natural large cardinal notion characterizing strong compactness cardinals for the equicardinality logic.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103583"},"PeriodicalIF":0.6,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143748012","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}
Pub Date : 2025-03-13DOI: 10.1016/j.apal.2025.103581
Martina Iannella , Alberto Marcone , Luca Motto Ros , Vadim Weinstein
Given a nonempty set of linear orders, we say that the linear order L is -convex embeddable into the linear order if it is possible to partition L into convex sets indexed by some element of which are isomorphic to convex subsets of ordered in the same way. This notion generalizes convex embeddability and (finite) piecewise convex embeddability (both studied in [13]), which are the special cases and . We focus mainly on the behavior of these relations on the set of countable linear orders, first characterizing when they are transitive, and hence a quasi-order. We then study these quasi-orders from a combinatorial point of view, and analyze their complexity with respect to Borel reducibility. Finally, we extend our analysis to uncountable linear orders.
{"title":"Piecewise convex embeddability on linear orders","authors":"Martina Iannella , Alberto Marcone , Luca Motto Ros , Vadim Weinstein","doi":"10.1016/j.apal.2025.103581","DOIUrl":"10.1016/j.apal.2025.103581","url":null,"abstract":"<div><div>Given a nonempty set <span><math><mi>L</mi></math></span> of linear orders, we say that the linear order <em>L</em> is <span><math><mi>L</mi></math></span>-convex embeddable into the linear order <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> if it is possible to partition <em>L</em> into convex sets indexed by some element of <span><math><mi>L</mi></math></span> which are isomorphic to convex subsets of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> ordered in the same way. This notion generalizes convex embeddability and (finite) piecewise convex embeddability (both studied in <span><span>[13]</span></span>), which are the special cases <span><math><mi>L</mi><mo>=</mo><mo>{</mo><mn>1</mn><mo>}</mo></math></span> and <span><math><mi>L</mi><mo>=</mo><mrow><mi>Fin</mi></mrow></math></span>. We focus mainly on the behavior of these relations on the set of countable linear orders, first characterizing when they are transitive, and hence a quasi-order. We then study these quasi-orders from a combinatorial point of view, and analyze their complexity with respect to Borel reducibility. Finally, we extend our analysis to uncountable linear orders.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103581"},"PeriodicalIF":0.6,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143769065","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}
Pub Date : 2025-03-12DOI: 10.1016/j.apal.2025.103582
Tristan van der Vlugt
We will give an overview of four families of cardinal characteristics defined on subspaces of the generalised Baire space , where κ is strongly inaccessible and . The considered families are bounded versions of the dominating, eventual difference, localisation and antilocalisation numbers, and their dual cardinals. We investigate parameters for which these cardinals are nontrivial and how the cardinals relate to each other and to other cardinals of the generalised Cichoń diagram. Finally we prove that different choices of parameters may lead to consistently distinct cardinals.
{"title":"Cardinal characteristics on bounded generalised Baire spaces","authors":"Tristan van der Vlugt","doi":"10.1016/j.apal.2025.103582","DOIUrl":"10.1016/j.apal.2025.103582","url":null,"abstract":"<div><div>We will give an overview of four families of cardinal characteristics defined on subspaces <span><math><msub><mrow><mo>∏</mo></mrow><mrow><mi>α</mi><mo>∈</mo><mi>κ</mi></mrow></msub><mi>b</mi><mo>(</mo><mi>α</mi><mo>)</mo></math></span> of the generalised Baire space <span><math><mmultiscripts><mrow><mi>κ</mi></mrow><mprescripts></mprescripts><none></none><mrow><mi>κ</mi></mrow></mmultiscripts></math></span>, where <em>κ</em> is strongly inaccessible and <span><math><mi>b</mi><mo>∈</mo><msup><mrow></mrow><mrow><mi>κ</mi></mrow></msup><mi>κ</mi></math></span>. The considered families are bounded versions of the dominating, eventual difference, localisation and antilocalisation numbers, and their dual cardinals. We investigate parameters for which these cardinals are nontrivial and how the cardinals relate to each other and to other cardinals of the generalised Cichoń diagram. Finally we prove that different choices of parameters may lead to consistently distinct cardinals.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 7","pages":"Article 103582"},"PeriodicalIF":0.6,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143686230","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}
Pub Date : 2025-03-10DOI: 10.1016/j.apal.2025.103579
Piotr Borodulin-Nadzieja , Jonathan Cancino-Manríquez , Adam Morawski
We answer in negative the problem if the existence of a P-measure implies the existence of a P-point. Namely, we show that if we add random reals to a certain ‘unique P-point’ model, then in the resulting model we will have a P-measure but not P-points. Also, we investigate the question if there is a P-measure in the Silver model. We show that rapid filters cannot be extended to a P-measure in the extension by ω product of Silver forcings and that in the model obtained by the countable support -iteration of countable product of Silver forcings there are no P-measures of countable Maharam type.
{"title":"P-measures in models without P-points","authors":"Piotr Borodulin-Nadzieja , Jonathan Cancino-Manríquez , Adam Morawski","doi":"10.1016/j.apal.2025.103579","DOIUrl":"10.1016/j.apal.2025.103579","url":null,"abstract":"<div><div>We answer in negative the problem if the existence of a P-measure implies the existence of a P-point. Namely, we show that if we add random reals to a certain ‘unique P-point’ model, then in the resulting model we will have a P-measure but not P-points. Also, we investigate the question if there is a P-measure in the Silver model. We show that rapid filters cannot be extended to a P-measure in the extension by <em>ω</em> product of Silver forcings and that in the model obtained by the countable support <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-iteration of countable product of Silver forcings there are no P-measures of countable Maharam type.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 7","pages":"Article 103579"},"PeriodicalIF":0.6,"publicationDate":"2025-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643186","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}
Pub Date : 2025-03-07DOI: 10.1016/j.apal.2025.103580
Wolfgang Rump
Main results on BL-algebras, including their classification in the finite case, are reconsidered and extended to a class of L-algebras X with prime factorization, including BL-algebras with ascending chain condition for its lattice. The weighted forest associated with a finite BL-algebra reappears as a canonical L-subalgebra of prime elements in the self-similar closure where is completely determined by its underlying poset (not necessarily a forest), while the weights are associated with existing powers of the prime elements in X. These invariants determine X within its self-similar closure . The three basic types of BL-algebras are related to concepts of L-algebras with further-reaching significance in quantum theory.
{"title":"A complete invariant system for noetherian BL-algebras and more general L-algebras","authors":"Wolfgang Rump","doi":"10.1016/j.apal.2025.103580","DOIUrl":"10.1016/j.apal.2025.103580","url":null,"abstract":"<div><div>Main results on <em>BL</em>-algebras, including their classification in the finite case, are reconsidered and extended to a class of <em>L</em>-algebras <em>X</em> with prime factorization, including <em>BL</em>-algebras with ascending chain condition for its lattice. The weighted forest associated with a finite <em>BL</em>-algebra reappears as a canonical <em>L</em>-subalgebra <span><math><mover><mrow><mi>P</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of prime elements in the self-similar closure <span><math><mi>S</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> where <span><math><mover><mrow><mi>P</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is completely determined by its underlying poset (not necessarily a forest), while the weights are associated with existing powers of the prime elements in <em>X</em>. These invariants determine <em>X</em> within its self-similar closure <span><math><mi>S</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>S</mi><mo>(</mo><mover><mrow><mi>P</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mi>X</mi><mo>)</mo><mo>)</mo></math></span>. The three basic types of <em>BL</em>-algebras are related to concepts of <em>L</em>-algebras with further-reaching significance in quantum theory.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 7","pages":"Article 103580"},"PeriodicalIF":0.6,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143620540","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}
Pub Date : 2025-03-04DOI: 10.1016/j.apal.2025.103570
Vera Fischer , L. Schembecker , David Schrittesser
We introduce the notion of a tight cofinitary group, which captures forcing indestructibility of maximal cofinitary groups for a long list of partial orders, including Cohen, Sacks, Miller, Miller partition forcing and Shelah's poset for diagonalizing maximal ideals. Introducing a new robust coding technique, we establish the relative consistency of alongside the existence of a -well-order of the reals and a co-analytic witness for .
{"title":"Tight cofinitary groups","authors":"Vera Fischer , L. Schembecker , David Schrittesser","doi":"10.1016/j.apal.2025.103570","DOIUrl":"10.1016/j.apal.2025.103570","url":null,"abstract":"<div><div>We introduce the notion of a tight cofinitary group, which captures forcing indestructibility of maximal cofinitary groups for a long list of partial orders, including Cohen, Sacks, Miller, Miller partition forcing and Shelah's poset for diagonalizing maximal ideals. Introducing a new robust coding technique, we establish the relative consistency of <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>=</mo><mi>d</mi><mo><</mo><mi>c</mi><mo>=</mo><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> alongside the existence of a <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-well-order of the reals and a co-analytic witness for <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103570"},"PeriodicalIF":0.6,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601168","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}
Pub Date : 2025-03-04DOI: 10.1016/j.apal.2025.103569
Morenikeji Neri
We explore the computational content of Kronecker's lemma via the proof-theoretic perspective of proof mining and utilise the resulting finitary variant of this fundamental result to provide new rates for the Strong Law of Large Numbers for random variables taking values in type p Banach spaces, which in particular are very uniform in the sense that they do not depend on the distribution of the random variables. Furthermore, we provide computability-theoretic arguments to demonstrate the ineffectiveness of Kronecker's lemma and investigate the result from the perspective of Reverse Mathematics. In addition, we demonstrate how this ineffectiveness from Kronecker's lemma trickles down to the Strong Law of Large Numbers by providing a construction that shows that computable rates of convergence are not always possible. Lastly, we demonstrate how Kronecker's lemma falls under a class of deterministic formulas whose solution to their Dialectica interpretation satisfies a continuity property and how, for such formulas, one obtains an upgrade principle that allows one to lift computational interpretations of deterministic results to quantitative results for their probabilistic analogue. This result generalises the previous work of the author and Pischke.
我们通过证明挖掘的证明论视角探索克罗内克两难的计算内容，并利用这一基本结果的有限变体，为在 p 型巴拿赫空间取值的随机变量的强大数定律提供新的速率，特别是在不依赖于随机变量分布的意义上，这种速率是非常均匀的。此外，我们还提供了可计算性理论论据来证明克罗内克∞的无效性，并从逆数学的角度研究了这一结果。此外，我们还提供了一个构造，说明可计算的收敛率并不总是可能的，以此证明克罗内克两难的无效性是如何向下渗透到大数强律的。最后，我们证明了克罗内克两难如何属于一类确定性公式，其辩证解释的解满足连续性属性，以及对于这类公式，我们如何获得一个升级原理，允许我们将确定性结果的计算解释提升为其概率类似的定量结果。这一结果概括了作者和皮施克之前的工作。
{"title":"A finitary Kronecker's lemma and large deviations in the strong law of large numbers on Banach spaces","authors":"Morenikeji Neri","doi":"10.1016/j.apal.2025.103569","DOIUrl":"10.1016/j.apal.2025.103569","url":null,"abstract":"<div><div>We explore the computational content of Kronecker's lemma via the proof-theoretic perspective of proof mining and utilise the resulting finitary variant of this fundamental result to provide new rates for the Strong Law of Large Numbers for random variables taking values in type <em>p</em> Banach spaces, which in particular are very uniform in the sense that they do not depend on the distribution of the random variables. Furthermore, we provide computability-theoretic arguments to demonstrate the ineffectiveness of Kronecker's lemma and investigate the result from the perspective of Reverse Mathematics. In addition, we demonstrate how this ineffectiveness from Kronecker's lemma trickles down to the Strong Law of Large Numbers by providing a construction that shows that computable rates of convergence are not always possible. Lastly, we demonstrate how Kronecker's lemma falls under a class of deterministic formulas whose solution to their Dialectica interpretation satisfies a continuity property and how, for such formulas, one obtains an upgrade principle that allows one to lift computational interpretations of deterministic results to quantitative results for their probabilistic analogue. This result generalises the previous work of the author and Pischke.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103569"},"PeriodicalIF":0.6,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562528","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}
Pub Date : 2025-03-03DOI: 10.1016/j.apal.2025.103566
V. Fischer , L. Schembecker
We introduce the notion of an arithmetical type of combinatorial family of reals, which serves to generalize different types of families such as mad families, maximal cofinitary groups, ultrafilter bases, splitting families and other similar types of families commonly studied in combinatorial set theory.
We then prove that every combinatorial family of reals of arithmetical type which is indestructible by the product of Sacks forcing is in fact universally Sacks-indestructible, i.e. it is indestructible by any countably supported iteration or product of Sacks-forcing of any length. Further, under we present a unified construction of universally Sacks-indestructible families for various arithmetical types of families. In particular we prove the existence of a universally Sacks-indestructible maximal cofinitary group under .
{"title":"Universally Sacks-indestructible combinatorial families of reals","authors":"V. Fischer , L. Schembecker","doi":"10.1016/j.apal.2025.103566","DOIUrl":"10.1016/j.apal.2025.103566","url":null,"abstract":"<div><div>We introduce the notion of an arithmetical type of combinatorial family of reals, which serves to generalize different types of families such as mad families, maximal cofinitary groups, ultrafilter bases, splitting families and other similar types of families commonly studied in combinatorial set theory.</div><div>We then prove that every combinatorial family of reals of arithmetical type which is indestructible by the product of Sacks forcing <span><math><msup><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span> is in fact universally Sacks-indestructible, i.e. it is indestructible by any countably supported iteration or product of Sacks-forcing of any length. Further, under <span><math><mi>CH</mi></math></span> we present a unified construction of universally Sacks-indestructible families for various arithmetical types of families. In particular we prove the existence of a universally Sacks-indestructible maximal cofinitary group under <span><math><mi>CH</mi></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103566"},"PeriodicalIF":0.6,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562527","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}
Pub Date : 2025-02-28DOI: 10.1016/j.apal.2025.103567
Miloš S. Kurilić
denotes the reduced power of a Boolean algebra , where Φ is the Fréchet filter on ω. We investigate iterated reduced powers ( and ) of collapsing algebras and our main intention is to classify the algebras , , up to isomorphism of their Boolean completions. In particular, assuming that SCH and hold, we show that for any cardinals and such that and we have , for each ; more precisely,
{"title":"Iterated reduced powers of collapsing algebras","authors":"Miloš S. Kurilić","doi":"10.1016/j.apal.2025.103567","DOIUrl":"10.1016/j.apal.2025.103567","url":null,"abstract":"<div><div><span><math><mrow><mi>rp</mi></mrow><mo>(</mo><mi>B</mi><mo>)</mo></math></span> denotes the reduced power <span><math><msup><mrow><mi>B</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>/</mo><mi>Φ</mi></math></span> of a Boolean algebra <span><math><mi>B</mi></math></span>, where Φ is the Fréchet filter on <em>ω</em>. We investigate iterated reduced powers (<span><math><msup><mrow><mi>rp</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><mi>B</mi><mo>)</mo><mo>=</mo><mi>B</mi></math></span> and <span><math><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>B</mi><mo>)</mo><mo>=</mo><mrow><mi>rp</mi></mrow><mo>(</mo><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>B</mi><mo>)</mo><mo>)</mo></math></span>) of collapsing algebras and our main intention is to classify the algebras <span><math><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mrow><mi>Col</mi></mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>κ</mi><mo>)</mo><mo>)</mo></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, up to isomorphism of their Boolean completions. In particular, assuming that SCH and <span><math><mi>h</mi><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> hold, we show that for any cardinals <span><math><mi>λ</mi><mo>≥</mo><mi>ω</mi></math></span> and <span><math><mi>κ</mi><mo>≥</mo><mn>2</mn></math></span> such that <span><math><mi>κ</mi><mi>λ</mi><mo>></mo><mi>ω</mi></math></span> and <span><math><mrow><mi>cf</mi></mrow><mo>(</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mi>c</mi></math></span> we have <span><math><mrow><mi>ro</mi></mrow><mo>(</mo><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mrow><mi>Col</mi></mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>κ</mi><mo>)</mo><mo>)</mo><mo>)</mo><mo>≅</mo><mrow><mi>Col</mi></mrow><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msup><mrow><mo>(</mo><msup><mrow><mi>κ</mi></mrow><mrow><mo><</mo><mi>λ</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>ω</mi></mrow></msup><mo>)</mo></math></span>, for each <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>; more precisely,<span><span><span><math><mrow><mi>ro</mi></mrow><mo>(</mo><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mrow><mi>Col</mi></mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>κ</mi><mo>)</mo><mo>)</mo><mo>)</mo><mo>≅</mo><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mi>Col</mi></mrow><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>c</mi><mo>)</mo><mo>,</mo><mspace></mspace></mtd><mtd><mtext> if </mtext><msup><mrow><mi>κ</mi></mrow><mrow><mo><</mo><mi>λ</mi></mrow></msup><mo>≤</mo><mi>c</mi><mo>;</mo></mtd></mtr><mtr><mtd><mrow><mi>Col</mi></mrow><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msup><mrow><mi>κ</mi></mrow><mrow><mo><</mo><mi>λ</mi></mrow></msup><mo>)</mo><mo>,</mo><mspace></mspa","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103567"},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143547846","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}
Pub Date : 2025-02-28DOI: 10.1016/j.apal.2025.103568
Anand Pillay
We adapt the notion from [7] and [2] of a (relatively) definable subset of when M is a saturated structure, to the case when M is atomic and strongly ω-homogeneous (over a set A). We discuss the existence and uniqueness of invariant measures on the Boolean algebra of definable subsets of . For example when T is stable, we have existence and uniqueness.
We also discuss the compatibility of our definability notions with definable Galois cohomology from [12] and differential Galois theory.
{"title":"Automorphism groups of prime models, and invariant measures","authors":"Anand Pillay","doi":"10.1016/j.apal.2025.103568","DOIUrl":"10.1016/j.apal.2025.103568","url":null,"abstract":"<div><div>We adapt the notion from <span><span>[7]</span></span> and <span><span>[2]</span></span> of a (relatively) definable subset of <span><math><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>M</mi><mo>)</mo></math></span> when <em>M</em> is a saturated structure, to the case <span><math><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>M</mi><mo>/</mo><mi>A</mi><mo>)</mo></math></span> when <em>M</em> is atomic and strongly <em>ω</em>-homogeneous (over a set <em>A</em>). We discuss the existence and uniqueness of invariant measures on the Boolean algebra of definable subsets of <span><math><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>M</mi><mo>/</mo><mi>A</mi><mo>)</mo></math></span>. For example when <em>T</em> is stable, we have existence and uniqueness.</div><div>We also discuss the compatibility of our definability notions with definable Galois cohomology from <span><span>[12]</span></span> and differential Galois theory.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103568"},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143547845","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}
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