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Generalization of Shapiro’s theorem to higher arities and noninjective notations Shapiro定理在更高精度和非射符号中的推广
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-09-14 DOI: 10.1007/s00153-022-00836-4
Dariusz Kalociński, Michał Wrocławski
{"title":"Generalization of Shapiro’s theorem to higher arities and noninjective notations","authors":"Dariusz Kalociński,&nbsp;Michał Wrocławski","doi":"10.1007/s00153-022-00836-4","DOIUrl":"10.1007/s00153-022-00836-4","url":null,"abstract":"<div><p>In the framework of Stewart Shapiro, computations are performed directly on strings of symbols (numerals) whose abstract numerical interpretation is determined by a notation. Shapiro showed that a total unary function (unary relation) on natural numbers is computable in every injective notation if and only if it is almost constant or almost identity function (finite or co-finite set). We obtain a syntactic generalization of this theorem, in terms of quantifier-free definability, for functions and relations relatively intrinsically computable on certain types of equivalence structures. We also characterize the class of relations and partial functions of arbitrary finite arities which are computable in every notation (be it injective or not). We consider the same question for notations in which certain equivalence relations are assumed to be computable. Finally, we discuss connections with a theorem by Ash, Knight, Manasse and Slaman which allow us to deduce some (but not all) of our results, based on quantifier elimination.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 1-2","pages":"257 - 288"},"PeriodicalIF":0.3,"publicationDate":"2022-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-022-00836-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45481859","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: 1
Combinatorial properties and dependent choice in symmetric extensions based on Lévy collapse 基于Lévy折叠的对称扩展中的组合性质和依赖选择
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-09-10 DOI: 10.1007/s00153-022-00845-3
Amitayu Banerjee
{"title":"Combinatorial properties and dependent choice in symmetric extensions based on Lévy collapse","authors":"Amitayu Banerjee","doi":"10.1007/s00153-022-00845-3","DOIUrl":"10.1007/s00153-022-00845-3","url":null,"abstract":"<div><p>We work with symmetric extensions based on Lévy collapse and extend a few results of Apter, Cody, and Koepke. We prove a conjecture of Dimitriou from her Ph.D. thesis. We also observe that if <i>V</i> is a model of <span>(textsf {ZFC})</span>, then <span>(textsf {DC}_{&lt;kappa })</span> can be preserved in the symmetric extension of <i>V</i> in terms of symmetric system <span>(langle {mathbb {P}},{mathcal {G}},{mathcal {F}}rangle )</span>, if <span>({mathbb {P}})</span> is <span>(kappa )</span>-distributive and <span>({mathcal {F}})</span> is <span>(kappa )</span>-complete. Further we observe that if <span>(delta &lt;kappa )</span> and <i>V</i> is a model of <span>(textsf {ZF}+textsf {DC}_{delta })</span>, then <span>(textsf {DC}_{delta })</span> can be preserved in the symmetric extension of <i>V</i> in terms of symmetric system <span>(langle {mathbb {P}},{mathcal {G}},{mathcal {F}}rangle )</span>, if <span>({mathbb {P}})</span> is (<span>(delta +1)</span>)-strategically closed and <span>({mathcal {F}})</span> is <span>(kappa )</span>-complete.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 3-4","pages":"369 - 399"},"PeriodicalIF":0.3,"publicationDate":"2022-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-022-00845-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48900514","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: 4
The additive structure of integers with the lower Wythoff sequence 具有下Wythoff序列的整数的加性结构
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-09-06 DOI: 10.1007/s00153-022-00846-2
Mohsen Khani, Afshin Zarei
{"title":"The additive structure of integers with the lower Wythoff sequence","authors":"Mohsen Khani,&nbsp;Afshin Zarei","doi":"10.1007/s00153-022-00846-2","DOIUrl":"10.1007/s00153-022-00846-2","url":null,"abstract":"<div><p>We have provided a model-theoretic proof for the decidability of the additive structure of integers together with the function <i>f</i> mapping <i>x</i> to <span>(lfloor varphi xrfloor )</span> where <span>(varphi )</span> is the golden ratio.\u0000\u0000\u0000\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 1-2","pages":"225 - 237"},"PeriodicalIF":0.3,"publicationDate":"2022-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-022-00846-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42102908","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: 2
On forcing over (L(mathbb {R})) 关于强迫 (L(mathbb {R}))
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-09-03 DOI: 10.1007/s00153-022-00844-4
Daniel W. Cunningham
{"title":"On forcing over (L(mathbb {R}))","authors":"Daniel W. Cunningham","doi":"10.1007/s00153-022-00844-4","DOIUrl":"10.1007/s00153-022-00844-4","url":null,"abstract":"<div><p>Given that <span>(L(mathbb {R})models {text {ZF}}+ {text {AD}}+{text {DC}})</span>, we present conditions under which one can generically add new elements to <span>(L(mathbb {R}))</span> and obtain a model of <span>({text {ZF}}+ {text {AD}}+{text {DC}})</span>. This work is motivated by the desire to identify the smallest cardinal <span>(kappa )</span> in <span>(L(mathbb {R}))</span> for which one can generically add a new subset <span>(gsubseteq kappa )</span> to <span>(L(mathbb {R}))</span> such that <span>(L(mathbb {R})(g)models {text {ZF}}+ {text {AD}}+{text {DC}})</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 3-4","pages":"359 - 367"},"PeriodicalIF":0.3,"publicationDate":"2022-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50007506","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
On forcing over L(R)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L(mathbb {R})$$end{document} On forcing over L(R)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L(mathbb {R})$$end{document}
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-09-03 DOI: 10.1007/s00153-022-00844-4
D. Cunningham
{"title":"On forcing over L(R)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L(mathbb {R})$$end{document}","authors":"D. Cunningham","doi":"10.1007/s00153-022-00844-4","DOIUrl":"https://doi.org/10.1007/s00153-022-00844-4","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 1","pages":"359 - 367"},"PeriodicalIF":0.3,"publicationDate":"2022-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52099079","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
Wellfoundedness proof with the maximal distinguished set 具有最大可分辨集的Wellfoundness证明
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-08-24 DOI: 10.1007/s00153-022-00840-8
Toshiyasu Arai
{"title":"Wellfoundedness proof with the maximal distinguished set","authors":"Toshiyasu Arai","doi":"10.1007/s00153-022-00840-8","DOIUrl":"10.1007/s00153-022-00840-8","url":null,"abstract":"<div><p>In Arai (An ordinal analysis of a single stable ordinal, submitted) it is shown that an ordinal <span>(sup _{N&lt;omega }psi _{varOmega _{1}}(varepsilon _{varOmega _{{mathbb {S}}+N}+1}))</span> is an upper bound for the proof-theoretic ordinal of a set theory <span>(mathsf {KP}ell ^{r}+(Mprec _{Sigma _{1}}V))</span>. In this paper we show that a second order arithmetic <span>(Sigma ^{1-}_{2}{mathrm {-CA}}+Pi ^{1}_{1}{mathrm {-CA}}_{0})</span> proves the wellfoundedness up to <span>(psi _{varOmega _{1}}(varepsilon _{varOmega _{{mathbb {S}}+N+1}}))</span> for each <i>N</i>. It is easy to interpret <span>(Sigma ^{1-}_{2}{mathrm {-CA}}+Pi ^{1}_{1}{mathrm {-CA}}_{0})</span> in <span>(mathsf {KP}ell ^{r}+(Mprec _{Sigma _{1}}V))</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 3-4","pages":"333 - 357"},"PeriodicalIF":0.3,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44664889","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: 2
Involutive Uninorm Logic with Fixed Point enjoys finite strong standard completeness 具有不动点的对合一致逻辑具有有限强标准完备性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-07-22 DOI: 10.1007/s00153-022-00839-1
Sándor Jenei
{"title":"Involutive Uninorm Logic with Fixed Point enjoys finite strong standard completeness","authors":"Sándor Jenei","doi":"10.1007/s00153-022-00839-1","DOIUrl":"10.1007/s00153-022-00839-1","url":null,"abstract":"<div><p>An algebraic proof is presented for the finite strong standard completeness of the Involutive Uninorm Logic with Fixed Point (<span>({{mathbf {IUL}}^{fp}})</span>). It may provide a first step towards settling the standard completeness problem for the Involutive Uninorm Logic (<span>({mathbf {IUL}})</span>, posed in G. Metcalfe, F. Montagna. (J Symb Log 72:834–864, 2007)) in an algebraic manner. The result is proved via an embedding theorem which is based on the structural description of the class of odd involutive FL<span>(_e)</span>-chains which have finitely many positive idempotent elements.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 1-2","pages":"67 - 86"},"PeriodicalIF":0.3,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-022-00839-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47378283","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: 2
Complexity of (Sigma ^0_n)-classifications for definable subsets 可定义子集的(Sigma ^0_n) -分类的复杂性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-07-20 DOI: 10.1007/s00153-022-00842-6
Svetlana Aleksandrova, Nikolay Bazhenov, Maxim Zubkov
{"title":"Complexity of (Sigma ^0_n)-classifications for definable subsets","authors":"Svetlana Aleksandrova,&nbsp;Nikolay Bazhenov,&nbsp;Maxim Zubkov","doi":"10.1007/s00153-022-00842-6","DOIUrl":"10.1007/s00153-022-00842-6","url":null,"abstract":"<div><p>For a non-zero natural number <i>n</i>, we work with finitary <span>(Sigma ^0_n)</span>-formulas <span>(psi (x))</span> without parameters. We consider computable structures <span>({mathcal {S}})</span> such that the domain of <span>({mathcal {S}})</span> has infinitely many <span>(Sigma ^0_n)</span>-definable subsets. Following Goncharov and Kogabaev, we say that an infinite list of <span>(Sigma ^0_n)</span>-formulas is a <span>(Sigma ^0_n)</span>-<i>classification</i> for <span>({mathcal {S}})</span> if the list enumerates all <span>(Sigma ^0_n)</span>-definable subsets of <span>({mathcal {S}})</span> without repetitions. We show that an arbitrary computable <span>({mathcal {S}})</span> always has a <span>({{mathbf {0}}}^{(n)})</span>-computable <span>(Sigma ^0_n)</span>-classification. On the other hand, we prove that this bound is sharp: we build a computable structure with no <span>({{mathbf {0}}}^{(n-1)})</span>-computable <span>(Sigma ^0_n)</span>-classifications.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 1-2","pages":"239 - 256"},"PeriodicalIF":0.3,"publicationDate":"2022-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50038891","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
Complexity of $$Sigma ^0_n$$ Σ n 0 -classifications for definable subsets $$Sigma^0_n$$∑n0-可定义子集分类的复杂性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-07-20 DOI: 10.1007/s00153-022-00842-6
S. Aleksandrova, N. Bazhenov, M. Zubkov
{"title":"Complexity of \u0000 \u0000 \u0000 \u0000 $$Sigma ^0_n$$\u0000 \u0000 \u0000 Σ\u0000 n\u0000 0\u0000 \u0000 \u0000 -classifications for definable subsets","authors":"S. Aleksandrova, N. Bazhenov, M. Zubkov","doi":"10.1007/s00153-022-00842-6","DOIUrl":"https://doi.org/10.1007/s00153-022-00842-6","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 1","pages":"239-256"},"PeriodicalIF":0.3,"publicationDate":"2022-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47638392","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
An ordinal-connection axiom as a weak form of global choice under the GCH GCH下作为全局选择弱形式的序数连接公理
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-07-13 DOI: 10.1007/s00153-022-00838-2
Rodrigo A. Freire, Peter Holy
{"title":"An ordinal-connection axiom as a weak form of global choice under the GCH","authors":"Rodrigo A. Freire,&nbsp;Peter Holy","doi":"10.1007/s00153-022-00838-2","DOIUrl":"10.1007/s00153-022-00838-2","url":null,"abstract":"<div><p>The minimal ordinal-connection axiom <span>(MOC)</span> was introduced by the first author in R. Freire. (South Am. J. Log. 2:347–359, 2016). We observe that <span>(MOC)</span> is equivalent to a number of statements on the existence of certain hierarchies on the universe, and that under global choice, <span>(MOC)</span> is in fact equivalent to the <span>({{,mathrm{GCH},}})</span>. Our main results then show that <span>(MOC)</span> corresponds to a weak version of global choice in models of the <span>({{,mathrm{GCH},}})</span>: it can fail in models of the <span>({{,mathrm{GCH},}})</span> without global choice, but also global choice can fail in models of <span>(MOC)</span>. \u0000\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 3-4","pages":"321 - 332"},"PeriodicalIF":0.3,"publicationDate":"2022-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49255186","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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