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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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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
Positive logics 积极的逻辑
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-07-09 DOI: 10.1007/s00153-022-00837-3
Saharon Shelah, Jouko Väänänen
{"title":"Positive logics","authors":"Saharon Shelah,&nbsp;Jouko Väänänen","doi":"10.1007/s00153-022-00837-3","DOIUrl":"10.1007/s00153-022-00837-3","url":null,"abstract":"<div><p>Lindström’s Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context of negation-less logics, <i>positive logics</i>, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-022-00837-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9113041","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
A few more dissimilarities between second-order arithmetic and set theory 二阶算术与集合论的几点不同
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-07-09 DOI: 10.1007/s00153-022-00829-3
Kentaro Fujimoto
{"title":"A few more dissimilarities between second-order arithmetic and set theory","authors":"Kentaro Fujimoto","doi":"10.1007/s00153-022-00829-3","DOIUrl":"10.1007/s00153-022-00829-3","url":null,"abstract":"<div><p>Second-order arithmetic and class theory are second-order theories of mathematical subjects of foundational importance, namely, arithmetic and set theory. Despite the similarity in appearance, there turned out to be significant mathematical dissimilarities between them. The present paper studies various principles in class theory, from such a comparative perspective between second-order arithmetic and class theory, and presents a few new dissimilarities between them.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-022-00829-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48622961","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
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