Annals of Pure and Applied Logic最新文献

{"title":"Searching problems above arithmetical transfinite recursion","authors":"Yudai Suzuki ,&nbsp;Keita Yokoyama","doi":"10.1016/j.apal.2024.103488","DOIUrl":"https://doi.org/10.1016/j.apal.2024.103488","url":null,"abstract":"<div><p>We investigate some Weihrauch problems between <span><math><msub><mrow><mi>ATR</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></msub></math></span>. We show that the fixed point theorem for monotone operators on the Cantor space (a weaker version of the Knaster-Tarski theorem) is not Weihrauch reducible to <span><math><msub><mrow><mi>ATR</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Furthermore, we introduce the <em>ω</em>-model reflection <span><math><msubsup><mrow><mi>ATR</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>rfn</mi></mrow></msubsup></math></span> of <span><math><msub><mrow><mi>ATR</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and show that it is an upper bound for problems provable from the axiomatic system <span><math><msub><mrow><mi>ATR</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> which are of the form <span><math><mo>∀</mo><mi>X</mi><mo>(</mo><mi>θ</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>→</mo><mo>∃</mo><mi>Y</mi><mi>η</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo><mo>)</mo></math></span> with arithmetical formulas <span><math><mi>θ</mi><mo>,</mo><mi>η</mi></math></span>. We also show that Weihrauch degrees of relativized least fixed point theorems for monotone operators on the Cantor space form a linear hierarchy between <span><math><msubsup><mrow><mi>ATR</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>rfn</mi></mrow></msubsup></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></mrow></msub></math></span>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103488"},"PeriodicalIF":0.6,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141540300","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}

{"title":"Decidability bounds for Presburger arithmetic extended by sine","authors":"","doi":"10.1016/j.apal.2024.103487","DOIUrl":"10.1016/j.apal.2024.103487","url":null,"abstract":"<div><p>We consider Presburger arithmetic extended by the sine function, call this extension sine-Presburger arithmetic (<strong>sin-PA</strong>), and systematically study decision problems for sets of sentences in <strong>sin-PA</strong>. In particular, we detail a decision algorithm for existential sin-PA sentences under assumption of Schanuel's conjecture. This procedure reduces decisions to the theory of the ordered additive group of real numbers extended by sine, which is decidable under Schanuel's conjecture. On the other hand, we prove that four alternating quantifier blocks suffice for undecidability of sin-PA sentences. To do so, we explicitly interpret the weak monadic second-order theory of the grid, which is undecidable, in <strong>sin-PA</strong>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103487"},"PeriodicalIF":0.6,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000915/pdfft?md5=8ec66c0193137f3b2153f14d4d1e4bed&pid=1-s2.0-S0168007224000915-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509201","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}

{"title":"From GTC to : Generating reset proof systems from cyclic proof systems","authors":"Graham E. Leigh,&nbsp;Dominik Wehr","doi":"10.1016/j.apal.2024.103485","DOIUrl":"10.1016/j.apal.2024.103485","url":null,"abstract":"<div><p>We consider cyclic proof systems in which derivations are graphs rather than trees. Such systems typically come with a condition that isolates which derivations are admitted as proofs, known as the <em>soundness condition</em>. This soundness condition frequently takes the form of either a <em>global trace</em> condition, a property dependent on all infinite paths in the proof-graph, or a <em>reset</em> condition, a ‘local’ condition depending on the simple cycles only which, as a result, is typically stable under more proof transformations.</p><p>In this article we present a general method for constructing cyclic proof systems with reset conditions from systems with global trace conditions. In contrast to previous approaches, this method of generation is entirely independent of logic's semantics, only relying on combinatorial aspects of the notion of ‘trace’ and ‘progress’. We apply this method to present reset proof systems for three cyclic proof systems from the literature: cyclic arithmetic, cyclic Gödel's T and cyclic tableaux for the modal <em>μ</em>-calculus.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103485"},"PeriodicalIF":0.6,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000897/pdfft?md5=3f6516f2a534f0fa710275ea2d71b171&pid=1-s2.0-S0168007224000897-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141402450","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}

{"title":"Posets of copies of countable ultrahomogeneous tournaments","authors":"Miloš S. Kurilić ,&nbsp;Stevo Todorčević","doi":"10.1016/j.apal.2024.103486","DOIUrl":"https://doi.org/10.1016/j.apal.2024.103486","url":null,"abstract":"&lt;div&gt;&lt;p&gt;The &lt;em&gt;poset of copies&lt;/em&gt; of a relational structure &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is the partial order &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;Y&lt;/mi&gt;&lt;mo&gt;≅&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and each similarity of such posets (e.g. isomorphism, forcing equivalence = isomorphism of Boolean completions, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;ro&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mi&gt;sq&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;) determines a classification of structures. Here we consider the structures from Lachlan's list of countable ultrahomogeneous tournaments: &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; (the rational line), &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; (the circular tournament), and &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; (the countable homogeneous universal tournament); as well as the ultrahomogeneous digraphs &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; from Cherlin's list.&lt;/p&gt;&lt;p&gt;If &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Rado&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; (resp. &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;) denotes the countable homogeneous universal graph (resp. &lt;em&gt;n&lt;/em&gt;-labeled linear order), it turns out that &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≅&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Rado&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and that &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; densely embeds in &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, for &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/p&gt;&lt;p&gt;Consequently, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≅&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;ro&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is the poset of perfect subsets of &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;em&gt;π&lt;/em&gt; an &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-name such that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103486"},"PeriodicalIF":0.8,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141325306","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}

{"title":"Around definable types in p-adically closed fields","authors":"Pablo Andújar Guerrero ,&nbsp;Will Johnson","doi":"10.1016/j.apal.2024.103484","DOIUrl":"https://doi.org/10.1016/j.apal.2024.103484","url":null,"abstract":"<div><p>We prove some technical results on definable types in <em>p</em>-adically closed fields, with consequences for definable groups and definable topological spaces. First, the code of a definable <em>n</em>-type (in the field sort) can be taken to be a real tuple (in the field sort) rather than an imaginary tuple (in the geometric sorts). Second, any definable type in the real or imaginary sorts is generated by a countable union of chains parameterized by the value group. Third, if <em>X</em> is an interpretable set, then the space of global definable types on <em>X</em> is strictly pro-interpretable, building off work of Cubides Kovacsics, Hils, and Ye <span>[7]</span>, <span>[8]</span>. Fourth, global definable types can be lifted (in a non-canonical way) along interpretable surjections. Fifth, if <em>G</em> is a definable group with definable f-generics (<em>dfg</em>), and <em>G</em> acts on a definable set <em>X</em>, then the quotient space <span><math><mi>X</mi><mo>/</mo><mi>G</mi></math></span> is definable, not just interpretable. This explains some phenomena observed by Pillay and Yao <span>[24]</span>. Lastly, we show that interpretable topological spaces satisfy analogues of first-countability and curve selection. Using this, we show that all reasonable notions of definable compactness agree on interpretable topological spaces, and that definable compactness is definable in families.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103484"},"PeriodicalIF":0.8,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141325305","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}

{"title":"Forcing axioms and the uniformization-property","authors":"Stefan Hoffelner","doi":"10.1016/j.apal.2024.103466","DOIUrl":"https://doi.org/10.1016/j.apal.2024.103466","url":null,"abstract":"<div><p>We show that there are models of <span><math><msub><mrow><mi>MA</mi></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></math></span> where the <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-uniformization property holds. Further we show that “<span><math><mi>BPFA</mi></math></span>+ <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is not inaccessible to reals” outright implies that the <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-uniformization property is true.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103466"},"PeriodicalIF":0.8,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000642/pdfft?md5=be051362a938ef048330838dac255f88&pid=1-s2.0-S0168007224000642-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141286509","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}

{"title":"Strong minimal pairs in the enumeration degrees","authors":"Josiah Jacobsen-Grocott","doi":"10.1016/j.apal.2024.103464","DOIUrl":"https://doi.org/10.1016/j.apal.2024.103464","url":null,"abstract":"<div><p>We prove that there are strong minimal pairs in the enumeration degrees and that the degrees of the left and right sides of strong minimal pairs include <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span> degrees, although it is unknown if there is a strong minimal pair in the <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span> enumeration degrees. We define a stronger type of minimal pair we call a strong super minimal pair, and show that there are none of these in the enumeration degrees, answering a question of Lempp et al. <span>[6]</span>. We leave open the question of the existence of a super minimal pair in the enumeration degrees.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103464"},"PeriodicalIF":0.8,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141286571","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}

{"title":"Finite undecidability in PAC and PRC fields","authors":"Brian Tyrrell","doi":"10.1016/j.apal.2024.103465","DOIUrl":"10.1016/j.apal.2024.103465","url":null,"abstract":"<div><p>A field <em>K</em> in a ring language <span><math><mi>L</mi></math></span> is <em>finitely undecidable</em> if <span><math><mtext>Cons</mtext><mo>(</mo><mi>Σ</mi><mo>)</mo></math></span> is undecidable for every nonempty finite <span><math><mi>Σ</mi><mo>⊆</mo><mtext>Th</mtext><mo>(</mo><mi>K</mi><mo>;</mo><mi>L</mi><mo>)</mo></math></span>. We adapt arguments originating with Cherlin-van den Dries-Macintyre/Ershov (for PAC fields) and Haran (for PRC fields) to prove all PAC and PRC fields are finitely undecidable. We describe the difficulties that arise in adapting the proof to P<em>p</em>C fields, and show no bounded P<em>p</em>C field is finitely axiomatisable. This work is drawn from the author's PhD thesis <span>[44, Chapter 4]</span>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103465"},"PeriodicalIF":0.8,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000630/pdfft?md5=4a33b42fff6d541d26261561103e7ddd&pid=1-s2.0-S0168007224000630-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141197022","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}

{"title":"Two-cardinal ideal operators and indescribability","authors":"Brent Cody ,&nbsp;Philip White","doi":"10.1016/j.apal.2024.103463","DOIUrl":"10.1016/j.apal.2024.103463","url":null,"abstract":"<div><p>A well-known version of Rowbottom's theorem for supercompactness ultrafilters leads naturally to notions of two-cardinal Ramseyness and corresponding normal ideals introduced herein. Generalizing results of Baumgartner, Feng and the first author, from the cardinal setting to the two-cardinal setting, we study hierarchies associated with a particular version of two-cardinal Ramseyness and a strong version of two-cardinal ineffability, as well as the relationships between these hierarchies and a natural notion of transfinite two-cardinal indescribability.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 8","pages":"Article 103463"},"PeriodicalIF":0.8,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000617/pdfft?md5=48f3bb6ab30a9410a6aa9d07709c915f&pid=1-s2.0-S0168007224000617-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061703","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}

{"title":"Fundamental sequences and fast-growing hierarchies for the Bachmann-Howard ordinal","authors":"David Fernández-Duque,&nbsp;Andreas Weiermann","doi":"10.1016/j.apal.2024.103455","DOIUrl":"10.1016/j.apal.2024.103455","url":null,"abstract":"<div><p>Hardy functions are defined by transfinite recursion and provide upper bounds for the growth rate of the provably total computable functions in various formal theories, making them an essential ingredient in many proofs of independence. Their definition is contingent on a choice of fundamental sequences, which approximate limits in a ‘canonical’ way. In order to ensure that these functions behave as expected, including the aforementioned unprovability results, these fundamental sequences must enjoy certain regularity properties.</p><p>In this article, we prove that Buchholz's system of fundamental sequences for the <em>ϑ</em> function enjoys such conditions, including the Bachmann property. We partially extend these results to variants of the <em>ϑ</em> function, including a version without addition for countable ordinals. We conclude that the Hardy functions based on these notation systems enjoy natural monotonicity properties and majorize all functions defined by primitive recursion along <span><math><mi>ϑ</mi><mo>(</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>Ω</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 8","pages":"Article 103455"},"PeriodicalIF":0.8,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000538/pdfft?md5=a9318d0df651509a7116d53069683110&pid=1-s2.0-S0168007224000538-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061705","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}