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Diagonalization as computation: Gödel sentence construction and a type-disciplined blueprint 作为计算的对角化：Gödel句子结构和类型规范的蓝图
IF 0.5 4区 数学
Archive for Mathematical Logic Pub Date : 2026-06-08 DOI: 10.1007/s00153-026-01021-7
Andrea Vestrucci
{"title":"Diagonalization as computation: Gödel sentence construction and a type-disciplined blueprint","authors":"Andrea Vestrucci","doi":"10.1007/s00153-026-01021-7","DOIUrl":"10.1007/s00153-026-01021-7","url":null,"abstract":"<div><p>We present a type-disciplined, compositional reconstruction of the diagonal strategy in Gödel’s first incompleteness theorem. The goal is not a new or “more direct” proof of incompleteness, but an explicit computational unpacking of the mechanism. The paper exhibits a concrete pipeline of code-transformers between primitive recursive syntactic domains, thereby avoiding well-definedness issues arising in Lindenbaum-quotient formulations. A central feature is the separation between a graph-mediated witness form of the Gödel sentence and a canonical normal form, connected by an internal normalization lemma formalizable in the base theory. The paper also presents the construction categorically in a category of primitive recursive code domains and morphisms, and extracts a general diagonal-flipping blueprint clarifying how type stratification distinguishes paradoxical from limitative diagonal arguments.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 6","pages":"761 - 799"},"PeriodicalIF":0.5,"publicationDate":"2026-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-026-01021-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149023256","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}
引用次数: 0
Relative model completeness of henselian valued fields with finite ramification and various value groups 具有有限分支和各种值群的henselian值域的相对模型完备性
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2026-05-25 DOI: 10.1007/s00153-026-01012-8
Anna De Mase
{"title":"Relative model completeness of henselian valued fields with finite ramification and various value groups","authors":"Anna De Mase","doi":"10.1007/s00153-026-01012-8","DOIUrl":"10.1007/s00153-026-01012-8","url":null,"abstract":"<div><p>We investigate the model completeness of the theory of a mixed characteristic henselian valued field with finite ramification relative to the model completeness of the residue field and value group. We address the case in which the valued field has a value group with finite spines, and the case in which the value group is elementarily equivalent to the infinite lexicographic sum of <span>(mathbb {Z})</span> with a minimal positive element. In both cases, we find a one-sorted language in which the theory of the valued field is model complete, if the theory of the residue field is model complete in the language of rings.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 5","pages":"569 - 591"},"PeriodicalIF":0.4,"publicationDate":"2026-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148196229","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}
引用次数: 0
Generalized Tukey reducibility between (sigma )-directed sets (sigma ) -有向集之间的广义Tukey可约性
IF 0.5 4区 数学
Archive for Mathematical Logic Pub Date : 2026-05-23 DOI: 10.1007/s00153-026-01019-1
Hiroshi Sakai, Toshimasa Tanno
{"title":"Generalized Tukey reducibility between (sigma )-directed sets","authors":"Hiroshi Sakai,&nbsp;Toshimasa Tanno","doi":"10.1007/s00153-026-01019-1","DOIUrl":"10.1007/s00153-026-01019-1","url":null,"abstract":"<div><p>We introduce a generalization of the Tukey reducibility, which we call the <i>pre-Tukey reducibility</i>. While the basics of the original Tukey reducibility heavily rely on the axiom of choice, the pre-Tukey reducibility works well in <span>(textsf{ZF})</span> (without the axiom of choice) to compare cofinal types of directed sets. In <span>(textsf{ZF})</span>, we show that two directed sets are pre-Tukey equivalent if and only if there exists a directed set into which both directed sets can be cofinally embedded. In this paper, we investigate the pre-Tukey reducibility between <span>(sigma )</span>-directed sets <span>((omega ^omega , le ^*), (mathcal {M}, subseteq ), (mathcal {N}, subseteq ), ([omega ^omega ]^omega , subseteq ) )</span> and <span>((omega _1, le ))</span> where <span>(le ^*)</span> denotes the dominating relation on <span>(omega ^omega )</span>, and <span>(mathcal {M} )</span> and <span>(mathcal {N})</span> denote the ideal of meager and null sets of <span>(2^omega )</span>, respectively. We show that under <span>(textsf{ZF}+ textsf{DC})</span> and certain assumptions on sets of reals, these directed sets have pairwise distinct cofinal types. The assumptions we consider hold in the Solovay model and in <span>(L(mathbb {R}))</span> satisfying the axiom of determinacy.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 6","pages":"701 - 718"},"PeriodicalIF":0.5,"publicationDate":"2026-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149023245","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}
引用次数: 0
Computability of a whitney extension 惠特尼扩展的可计算性
IF 0.5 4区 数学
Archive for Mathematical Logic Pub Date : 2026-05-17 DOI: 10.1007/s00153-026-01020-8
Andrea Brun, Guido Gherardi, Alberto Marcone
{"title":"Computability of a whitney extension","authors":"Andrea Brun,&nbsp;Guido Gherardi,&nbsp;Alberto Marcone","doi":"10.1007/s00153-026-01020-8","DOIUrl":"10.1007/s00153-026-01020-8","url":null,"abstract":"<div><p>We prove the computability of a version of Whitney Extension, when the input is suitably represented. More specifically, if <span>(F subseteq mathbb {R}^n)</span> is a closed set represented so that the distance function <span>(x mapsto d(x,F))</span> can be computed, and <span>((f^{(bar{k})})_{|bar{k}| le m})</span> is a Whitney jet of order <i>m</i> on <i>F</i>, then we can compute <span>(g in C^{m}(mathbb {R}^n))</span> such that <i>g</i> and its partial derivatives coincide on <i>F</i> with the corresponding functions of <span>((f^{(bar{k})})_{|bar{k}| le m})</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 6","pages":"719 - 760"},"PeriodicalIF":0.5,"publicationDate":"2026-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149023253","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}
引用次数: 0
Game-theoretic variants of splitting number 分裂数的博弈论变体
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2026-05-12 DOI: 10.1007/s00153-026-01015-5
Jorge Antonio Cruz Chapital, Tatsuya Goto, Yusuke Hayashi, Takashi Yamazoe
{"title":"Game-theoretic variants of splitting number","authors":"Jorge Antonio Cruz Chapital,&nbsp;Tatsuya Goto,&nbsp;Yusuke Hayashi,&nbsp;Takashi Yamazoe","doi":"10.1007/s00153-026-01015-5","DOIUrl":"10.1007/s00153-026-01015-5","url":null,"abstract":"<div><p>We consider combining the definition of a cardinal invariant and the notion of an infinite game. We focus on the splitting number <span>(mathfrak {s})</span> since the corresponding cardinal invariants behave in an interesting way. We introduce three kinds of games as reasonable realizations of the combination of the notions of splitting and infinite games. Then, we consider two cardinal invariants for each game, so we define six numbers. We prove that three of them are equal to the size of the continuum <span>(mathfrak {c})</span> and one of them is equal to the <span>(sigma )</span>-splitting number <span>(mathfrak {s}_sigma )</span>, which is defined as the minimum size of a <span>(sigma )</span>-splitting family. On the other hand, we show that the remaining two numbers are consistently different from <span>(mathfrak {c})</span>, <span>(mathfrak {s})</span> and <span>(mathfrak {s}_sigma )</span>. Moreover, though the two numbers share almost the same rule of the game, we prove that they can take distinct values from each other, and hence the slight difference of the rule is actually crucial in this sense.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 5","pages":"645 - 663"},"PeriodicalIF":0.4,"publicationDate":"2026-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148196230","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}
引用次数: 0
Binary choice games and arithmetical comprehension 二选游戏和算术理解
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2026-05-09 DOI: 10.1007/s00153-026-01016-4
J. P. Aguilera, T. Kouptchinsky
{"title":"Binary choice games and arithmetical comprehension","authors":"J. P. Aguilera,&nbsp;T. Kouptchinsky","doi":"10.1007/s00153-026-01016-4","DOIUrl":"10.1007/s00153-026-01016-4","url":null,"abstract":"<div><p>We prove that Arithmetical Comprehension is equivalent to the determinacy of all clopen integer games in which each player has at most two moves per turn.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 5","pages":"665 - 674"},"PeriodicalIF":0.4,"publicationDate":"2026-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-026-01016-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148196227","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}
引用次数: 0
There may be exactly n Q-points 可能正好有n个q点
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2026-05-09 DOI: 10.1007/s00153-026-01014-6
Lorenz Halbeisen, Silvan Horvath, Tan Özalp
{"title":"There may be exactly n Q-points","authors":"Lorenz Halbeisen,&nbsp;Silvan Horvath,&nbsp;Tan Özalp","doi":"10.1007/s00153-026-01014-6","DOIUrl":"10.1007/s00153-026-01014-6","url":null,"abstract":"<div><p>We generalize the main result of [13] and show the consistency of the statement “There are exactly <i>n</i> <i>Q</i>-points up to isomorphism\" for any finite <i>n</i>. Furthermore, we show that the above statement for <span>(n=2)</span> can alternatively be obtained by a length-<span>(omega _2)</span> countable support iteration of Matet-Mathias forcing restricted to a Matet-adequate family.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 5","pages":"631 - 643"},"PeriodicalIF":0.4,"publicationDate":"2026-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-026-01014-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148196231","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}
引用次数: 0
Reinhardt cardinals and eventually dominating functions 莱因哈特基数和支配函数
IF 0.5 4区 数学
Archive for Mathematical Logic Pub Date : 2026-05-04 DOI: 10.1007/s00153-026-01018-2
Marwan Salam Mohammd
{"title":"Reinhardt cardinals and eventually dominating functions","authors":"Marwan Salam Mohammd","doi":"10.1007/s00153-026-01018-2","DOIUrl":"10.1007/s00153-026-01018-2","url":null,"abstract":"<div><p>We study structural consequences of the existence of a Reinhardt embedding (an embedding of the set-theoretic universe into itself). First, we prove that if delta is a regular cardinal greater than the critical point of the embedding and is fixed by it, then no function from delta to delta in the range of the embedding eventually dominates the embedding restricted to delta. Assuming the existence of such an embedding, we use this to obtain elementary embeddings of initial segments of the universe into themselves exhibiting a local form of extendibility. We then refine results of Gabriel Goldberg on almost supercompact cardinals by proving that certain such cardinals cannot be regular. Finally, we use these ideas to give an alternative proof of Kunen’s Inconsistency Theorem.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 6","pages":"691 - 699"},"PeriodicalIF":0.5,"publicationDate":"2026-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149023252","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}
引用次数: 0
Correspondence of Contradictions in the Constructive Connexive Calculus C 构造连通演算中矛盾的对应性C
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2026-05-02 DOI: 10.1007/s00153-026-01013-7
Satoru Niki
{"title":"Correspondence of Contradictions in the Constructive Connexive Calculus C","authors":"Satoru Niki","doi":"10.1007/s00153-026-01013-7","DOIUrl":"10.1007/s00153-026-01013-7","url":null,"abstract":"<div><p>The calculus <b>C</b> was introduced by H. Wansing as a constructive logic with strong negation. In addition, <b>C</b> validates the theses of connexive logic that are attributed to Aristotle and Boethius. A further remarkable property of <b>C</b> is that it is a non-trivial but negation inconsistent system: it has a formula and its negation as theorems. From a bilateralist-minded perspective, such a contradiction can be seen as the existence of both a verification and a falsification of one and the same formula. Relatedly, it has been noted by Wansing that there seems to be a kind of correspondence between these two types of derivations when it comes to a proof of contradiction. Following this observation, we attempt in this paper to introduce a precise notion for such a correspondence. We thence establish that this correspondence obtains in propositional and first-order versions of <b>C</b>, via formulations of suitable sequent and tableau calculi.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 5","pages":"593 - 629"},"PeriodicalIF":0.4,"publicationDate":"2026-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-026-01013-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148196228","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}
引用次数: 0
On the (Sigma ^0_1)-fragments of the Kreisel-Putnam axiom and two variants of the linearity axiom in intuitionistic arithmetic and analysis 论直觉算术与分析中Kreisel-Putnam公理的(Sigma ^0_1) -片段和线性公理的两个变体
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2026-04-22 DOI: 10.1007/s00153-026-01017-3
Makoto Fujiwara
{"title":"On the (Sigma ^0_1)-fragments of the Kreisel-Putnam axiom and two variants of the linearity axiom in intuitionistic arithmetic and analysis","authors":"Makoto Fujiwara","doi":"10.1007/s00153-026-01017-3","DOIUrl":"10.1007/s00153-026-01017-3","url":null,"abstract":"<div><p>We study the <span>(Sigma ^0_1)</span>-fragment of the Kreisel-Putnam axiom and its <span>(Sigma ^0_n)</span>-extensions in the context of intuitionistic arithmetic and analysis. Among other things, we show that the <span>(Sigma ^0_1)</span>-fragment of the Kreisel-Putnam axiom is exactly the principle that fills the gap between the lesser limited principle of omniscience and the disjunctive Markov principle in constructive reverse mathematics. In addition, we introduce two variants of the linearity axiom, which are related to the Kreisel-Putnam axiom, and show that the <span>(Sigma ^0_1)</span>-fragments of these three axioms do not lie in any previously known layers in the arithmetical hierarchy of logical axioms.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 5","pages":"675 - 690"},"PeriodicalIF":0.4,"publicationDate":"2026-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148196226","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}
引用次数: 0
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