{"title":"A finite algebraic target theorem for Kreisel’s conjecture","authors":"Mario Piazza","doi":"10.1007/s00153-026-01029-z","DOIUrl":"10.1007/s00153-026-01029-z","url":null,"abstract":"<div><p>Friedman’s Problem 34, attributed there to Kreisel, states that for Peano arithmetic formalized precisely as in Kleene, a uniform bound on the proof lengths of all numeral instances <span>(A(bar{n}))</span> entails the provability of the universal closure <span>(forall x,A(x))</span>. We refer to this statement as Kreisel’s conjecture (KC). Parikh proved the corresponding implication for a relational presentation of Peano arithmetic. Bounded proof analyses have a finite automata-theoretic shadow: suppressing formula instantiations leaves a finite partial automaton whose states record only which schematic continuations remain possible. We characterize exactly when the map sending a word to the idempotent power of its transformation factors through a finite semilattice. The finite criterion separates a failure of commutativity among the idempotents associated with the generators from one that first appears for a composite word. The latter always has a witness whose length is bounded by the size of the transition monoid. In the positive case, the idempotent-power map retracts the transition monoid onto its full set of idempotents. Parikh’s case changes the picture. For the prefix automata of bounded proof trees, already before minimization every nonempty word has the empty partial map as its idempotent power, and the nonidentity transformations form a nilpotent ideal. Yet the universal implication still holds. What Parikh’s proof retains is not continuation behaviour but realization: which analyses prove which numeral instances. The resulting realization profiles determine the coarsest finite quotient of <span>(mathbb {N})</span> retaining every realization set. In Parikh’s presentation, the resulting equivalence relation and its classes are effectively definable in Presburger arithmetic, and the assignment sending each numeral to its realization profile is effectively ultimately periodic. The comparison separates exact algebraic compression of continuations from the arithmetic information needed for the passage to the universal closure.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 7","pages":"905 - 924"},"PeriodicalIF":0.5,"publicationDate":"2026-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-026-01029-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149176902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Unravelling cyclic first-order arithmetic","authors":"Graham E. Leigh, Dominik Wehr","doi":"10.1007/s00153-026-01030-6","DOIUrl":"10.1007/s00153-026-01030-6","url":null,"abstract":"<div><p>Cyclic proof systems for Heyting and Peano arithmetic eschew induction axioms by accepting proofs which are finite graphs rather than trees. Proving that such a cyclic proof system coincides with its more conventional variants is often difficult: previous proofs in the literature rely on intricate arithmetisations of the metamathematics of the cyclic proof systems. In this article, we present a simple and direct embedding of cyclic proofs for Heyting and Peano arithmetic into purely inductive, i.e. ‘finitary’, proofs by adapting a translation introduced by Sprenger and Dam for a cyclic proof system of <span>(mu textrm{FOL})</span> with explicit ordinal approximations. We adapt their method to obtain <span>(textrm{C}Pi _n subseteq textrm{I}Pi _{n + 1})</span>, extending Das’ result relating the traditional notion of inductive complexity to a similar cyclic notion. As part of the embedding we present a novel representation of cyclic proofs as a labelled sequent calculus which we call stack-controlled proofs.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 7","pages":"925 - 954"},"PeriodicalIF":0.5,"publicationDate":"2026-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-026-01030-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149176847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fragments of some subintuitionistic logics","authors":"Fatemeh Shirmohammadzadeh Maleki, Dick de Jongh","doi":"10.1007/s00153-026-01031-5","DOIUrl":"10.1007/s00153-026-01031-5","url":null,"abstract":"<div><p>In this paper we study, for a broad range of known subintuitionistic logics, their implicational fragment <span>(L_{[rightarrow ]})</span> and, in selected cases, the implication-conjunction fragment <span>(L_{[rightarrow ,wedge ]})</span>. For each logic under consideration we provide a Hilbert-style axiomatization of the corresponding fragment and establish strong completeness with respect to the appropriate (Kripke or neighborhood) semantics. This gives a systematic picture of implicational principles below <span>(textsf{IPC})</span> across the standard extensions of <span>(textsf{F})</span> and <span>(textsf{WF})</span>.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 7","pages":"955 - 978"},"PeriodicalIF":0.5,"publicationDate":"2026-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149176581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Infinitary negative translations and Glivenko logic","authors":"Giacomo Bartoli, Giulio Fellin, Matteo Tesi","doi":"10.1007/s00153-026-01028-0","DOIUrl":"10.1007/s00153-026-01028-0","url":null,"abstract":"<div><p>We study infinitary intuitionistic logic by employing both syntactic and semantic methods. First, we introduce a natural deduction system for infinitary predicate logic and study some of its basic properties. We then extend neighbourhood semantics to this setting, providing a soundness and completeness theorem for this system. Building on these tools, we derive three new conservation results connecting classical and constructive reasoning. We prove an infinitary version of Barr’s theorem using neighbourhood semantics; we define an infinitary Gentzen-style negative translation and show it yields a proper embedding of classical logic into minimal logic; and we establish an infinitary Glivenko theorem, identifying the Glivenko logic as the least extension of minimal logic validating Glivenko’s theorem. Finally, we show that the Glivenko logic is distinct from the other systems considered, thereby clarifying its place within the landscape of constructive infinitary logics.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 7","pages":"873 - 904"},"PeriodicalIF":0.5,"publicationDate":"2026-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-026-01028-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149176841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Homogeneity and orders between ideals","authors":"Krzysztof Kowitz, Jacek Tryba","doi":"10.1007/s00153-026-01032-4","DOIUrl":"10.1007/s00153-026-01032-4","url":null,"abstract":"<div><p>We consider several known ideals and check whether they are homogeneous. We also verify whether selected ideals are comparable in various orders, in particular Rudin-Blass, Rudin-Keisler and Katětov orders. As a consequence, we show that consistently there exists a Ramsey (differentially compact) space that is not a van der Waerden space – which is a partial answer to [13, Question 16.6] (positive answer to [37, Question 2] and partial answer to [48, Question 4.2.2]). We also prove the consistency of the existence of a differentially compact space that is not a Folkman space – which is a positive answer to [37, Question 1]. Moreover, we show that the concepts of a <span>(hat{P}^+)</span>-ideal and an indecomposable ideal are equivalent.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 7","pages":"979 - 1002"},"PeriodicalIF":0.5,"publicationDate":"2026-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149176559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Intuitionistic logic of a binary modality","authors":"Michael Kaminski","doi":"10.1007/s00153-026-01025-3","DOIUrl":"10.1007/s00153-026-01025-3","url":null,"abstract":"<div><p>We define the <i>intuitionistic</i> logic of a binary modality, present its semantics, and prove the completeness theorem.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 7","pages":"825 - 844"},"PeriodicalIF":0.5,"publicationDate":"2026-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-026-01025-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149176754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Canonical reconstruction and forcing absoluteness of standard structures","authors":"Tomasz Kania","doi":"10.1007/s00153-026-01026-2","DOIUrl":"10.1007/s00153-026-01026-2","url":null,"abstract":"<div><p>We isolate a simple preservation principle governing when it is absolute, between transitive models of set theory, that a given algebraic or topological-algebraic structure has a <i>standard form</i> <i>F</i>(<i>X</i>) indexed by a set <i>X</i>. The principle is: if the index <i>X</i> (or a proxy for it) can be recovered from <i>F</i>(<i>X</i>) by a uniform definable construction, then the class of structures isomorphic to some <i>F</i>(<i>X</i>) is downward absolute from forcing extensions. Answering a question raised by Noah Schweber, we deduce in particular that no group that fails to be a full symmetric group in the ground model can become one after forcing; the result holds already in <span>(textsf{ZF})</span>. The same mechanism applies to full transformation monoids, powerset Boolean algebras, full relation algebras, full clones, full partition lattices, products <span>(R^X)</span> of finitely generated centrally indecomposable rings, the commutative <span>(C^*)</span>-algebras <span>(ell _infty (X))</span> and <span>(c_0(X))</span>, full endomorphism rings, the operator algebras <span>(mathcal {B}(H))</span> and <span>(mathcal {K}(H))</span>, and <span>(ell _1(X))</span> as a real Banach lattice. In the motivating symmetric-group case, the same reconstruction gives more than descent: it yields a uniform <span>(Pi ^1_1)</span> definition of fullness over transitive <span>(textsf{ZF})</span>-models. We then exhibit clean torsor obstructions, in the standard symmetric-model situation: <i>finite covers</i> <span>(Y times n)</span> already separate <span>(textsf{ZF})</span>-failure from <span>(textsf{ZFC})</span>-descent without any completeness caveat, and the finite-support normed space <span>(c_{00}(I))</span> provides the analogous Banach example. Bare-Banach-space isomorphism with <span>(ell _1(Gamma ))</span> exhibits a genuine <span>(textsf{ZFC})</span>-descent. We conclude with the corresponding, relative, obstructions to <span>(Pi ^1_1)</span>-definability of standardness over transitive <span>(textsf{ZF})</span>-models.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 7","pages":"845 - 871"},"PeriodicalIF":0.5,"publicationDate":"2026-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-026-01026-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149176871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On some subtheories of strong dependent choice","authors":"Juan P. Aguilera, Yudai Suzuki, Keita Yokoyama","doi":"10.1007/s00153-026-01023-5","DOIUrl":"10.1007/s00153-026-01023-5","url":null,"abstract":"<div><p>In this paper, we give characterizations of the set of <span>(Pi ^1_{e+2})</span>-consequences, <span>(Sigma ^1_{e+1})</span>-consequences and <span>(textsf{B}(Pi ^1_{e+1}))</span>-consequences of the axiomatic system of strong dependent choice for <span>(Sigma ^1_i)</span> formulas <span>(Sigma ^1_i-textsf{SDC}_0)</span> for <span>(e < i)</span>. Here, <span>(textsf{B}(Gamma ))</span> denotes the set generated by <span>(wedge ,vee ,lnot )</span> starting from <span>(Gamma )</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 6","pages":"809 - 822"},"PeriodicalIF":0.5,"publicationDate":"2026-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149023254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Correction: Diagonalization as computation: Gödel sentence construction and a type-disciplined blueprint","authors":"Andrea Vestrucci","doi":"10.1007/s00153-026-01024-4","DOIUrl":"10.1007/s00153-026-01024-4","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 6","pages":"823 - 824"},"PeriodicalIF":0.5,"publicationDate":"2026-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-026-01024-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149023255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A further application of Stern absoluteness","authors":"Itay Neeman","doi":"10.1007/s00153-026-01022-6","DOIUrl":"10.1007/s00153-026-01022-6","url":null,"abstract":"<div><p>We prove, in <span>ZFC</span>, that certain <span>(varvec{Sigma }^1_2)</span> functions cannot injectively embed <span>(omega _1)</span> into a Borel class of fixed countable rank. This had been proved under determinacy or large cardinals by Harrington and Hjorth for all <span>(varvec{Sigma }^1_2)</span> functions. Our contribution is to identify conditions under which the determinacy and large cardinal assumptions can be removed. These conditions are sufficient for a recent use of the non-existence of <span>(varvec{Sigma }^1_2)</span> injections of <span>(omega _1)</span> into Borel classes by Day and Marks.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 6","pages":"801 - 807"},"PeriodicalIF":0.5,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149023246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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