Journal of Mathematical Logic最新文献

{"title":"The descriptive complexity of the set of Poisson generic numbers","authors":"Verónica Becher, Stephen Jackson, Dominik Kwietniak, Bill Mance","doi":"10.1142/s0219061324500193","DOIUrl":"https://doi.org/10.1142/s0219061324500193","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>b</mi><mo>≥</mo><mn>2</mn></math></span><span></span> be an integer. We show that the set of real numbers that are Poisson generic in base <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>b</mi></math></span><span></span> is <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mstyle><mtext mathvariant=\"normal\">Π</mtext></mstyle></mrow><mrow><mn>3</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span><span></span>-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>b</mi></math></span><span></span> and not Poisson generic in base <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>b</mi></math></span><span></span> is complete for the class given by the differences between <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mstyle><mtext mathvariant=\"normal\">Π</mtext></mstyle></mrow><mrow><mn>3</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span><span></span> sets. We also show that the effective versions of these results hold in the effective Borel hierarchy.</p>","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The equivalence of Axiom (*)+ and Axiom (*)++","authors":"W. Woodin","doi":"10.1142/s021906132450020x","DOIUrl":"https://doi.org/10.1142/s021906132450020x","url":null,"abstract":"","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140352942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-Galvin filters","authors":"Tom Benhamou, Shimon Garti, Moti Gitik, Alejandro Poveda","doi":"10.1142/s021906132450017x","DOIUrl":"https://doi.org/10.1142/s021906132450017x","url":null,"abstract":"<p>We address the question of consistency strength of certain filters and ultrafilters which fail to satisfy the Galvin property. We answer questions [Benhamou and Gitik, <i>Ann. Pure Appl. Logic</i><b>173</b> (2022) 103107; Questions 7.8, 7.9], [Benhamou <i>et al.</i>, <i>J. Lond. Math. Soc.</i><b>108</b>(1) (2023) 190–237; Question 5] and improve theorem [Benhamou <i>et al.</i>, <i>J. Lond. Math. Soc.</i><b>108</b>(1) (2023) 190–237; Theorem 2.3].</p>","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140314272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rings of finite Morley rank without the canonical base property","authors":"Michael Loesch, Daniel Palacín","doi":"10.1142/s0219061324500168","DOIUrl":"https://doi.org/10.1142/s0219061324500168","url":null,"abstract":"<p>We present numerous natural algebraic examples without the so-called Canonical Base Property (CBP). We prove that every commutative unitary ring of finite Morley rank without finite-index proper ideals satisfies the CBP if and only if it is a field, a ring of positive characteristic or a finite direct product of these. In addition, we construct a CM-trivial commutative local ring with a finite residue field without the CBP. Furthermore, we also show that finite-dimensional non-associative algebras over an algebraically closed field of characteristic <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mn>0</mn></math></span><span></span> give rise to triangular rings without the CBP. This also applies to Baudisch’s <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn></math></span><span></span>-step nilpotent Lie algebras, which yields the existence of a <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn></math></span><span></span>-step nilpotent group of finite Morley rank whose theory, in the pure language of groups, is CM-trivial and does not satisfy the CBP.</p>","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the consistency of ZF with an elementary embedding from Vλ+2 into Vλ+2","authors":"Farmer Schlutzenberg","doi":"10.1142/s0219061324500132","DOIUrl":"https://doi.org/10.1142/s0219061324500132","url":null,"abstract":"<p>According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>λ</mi></math></span><span></span> and nontrivial elementary embedding <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>j</mi><mo>:</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>λ</mi><mo stretchy=\"false\">+</mo><mn>2</mn></mrow></msub><mo>→</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>λ</mi><mo stretchy=\"false\">+</mo><mn>2</mn></mrow></msub></math></span><span></span>. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone is has been discovered.</p><p><span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>I</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>λ</mi></mrow></msub></math></span><span></span> is the assertion, introduced by Hugh Woodin, that <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>λ</mi></math></span><span></span> is an ordinal and there is an elementary embedding <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>j</mi><mo>:</mo><mi>L</mi><mo stretchy=\"false\">(</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>λ</mi><mo stretchy=\"false\">+</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo><mo>→</mo><mi>L</mi><mo stretchy=\"false\">(</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>λ</mi><mo stretchy=\"false\">+</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> with critical point <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mo>&lt;</mo><mi>λ</mi></math></span><span></span>. And <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>I</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span><span></span> asserts that <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>I</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>λ</mi></mrow></msub></math></span><span></span> holds for some <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>λ</mi></math></span><span></span>. The axiom <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>I</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span><span></span> is one of the strongest large cardinals not known to be inconsistent with AC. It is usually studied assuming ZFC in the full universe <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><mi>V</mi></math></span><span></span> (in which case <span><math altimg=\"eq-00015.gif\" display=\"inline\" overflow=\"scroll\"><mi>λ</mi></math></span><span></span> must be a limit ordinal), but we assume only ZF.</p><p>We prove, assuming ZF <span><math altimg=\"eq-00016.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">+</mo></math></span><span></span><span><math altimg=\"eq-00017.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>I</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>λ</mi></mrow></msub></math></span><span></span><span><","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The mouse set theorem just past projective","authors":"Mitch Rudominer","doi":"10.1142/s0219061324500144","DOIUrl":"https://doi.org/10.1142/s0219061324500144","url":null,"abstract":"<p>We identify a particular mouse, <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>M</mi></mrow><mrow><mstyle><mtext mathvariant=\"normal\">ld</mtext></mstyle></mrow></msup></math></span><span></span>, the minimal ladder mouse, that sits in the mouse order just past <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>♯</mi></mrow></msubsup></math></span><span></span> for all <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span>, and we show that <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>ℝ</mi><mspace width=\".17em\"></mspace><mo stretchy=\"false\">∩</mo><mspace width=\".17em\"></mspace><msup><mrow><mi>M</mi></mrow><mrow><mstyle><mtext mathvariant=\"normal\">ld</mtext></mstyle></mrow></msup><mo>=</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>ω</mi><mo stretchy=\"false\">+</mo><mn>1</mn></mrow></msub></math></span><span></span>, the set of reals that are <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mi mathvariant=\"normal\">Δ</mi></mrow><mrow><mi>ω</mi><mo stretchy=\"false\">+</mo><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span><span></span> in a countable ordinal. Thus <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>Q</mi></mrow><mrow><mi>ω</mi><mo stretchy=\"false\">+</mo><mn>1</mn></mrow></msub></math></span><span></span> is a mouse set.</p><p>This is analogous to the fact that <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>ℝ</mi><mspace width=\".17em\"></mspace><mo stretchy=\"false\">∩</mo><mspace width=\".17em\"></mspace><msubsup><mrow><mi>M</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>♯</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span><span></span> where <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mi>M</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>♯</mi></mrow></msubsup></math></span><span></span> is the sharp for the minimal inner model with a Woodin cardinal, and <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>Q</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span><span></span> is the set of reals that are <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mi mathvariant=\"normal\">Δ</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span><span></span> in a countable ordinal.</p><p>More generally <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>ℝ</mi><mspace width=\".17em\"></mspace><mo stretchy=\"false\">∩</mo><mspace width=\".17em\"></mspace><msubsup><mrow><mi>M</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo stretchy=\"false\">+</mo><mn>1</mn></mrow><mrow><mi>♯</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo stre","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On ω-Strongly Measurable Cardinals in ℙmax Extensions","authors":"Navin Aksornthong, Takehiko Gappo, James Holland, Grigor Sargsyan","doi":"10.1142/s0219061324500181","DOIUrl":"https://doi.org/10.1142/s0219061324500181","url":null,"abstract":"","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140257248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Martin's conjecture for regressive functions on the hyperarithmetic degrees","authors":"P. Lutz","doi":"10.1142/s0219061324500156","DOIUrl":"https://doi.org/10.1142/s0219061324500156","url":null,"abstract":"We answer a question of Slaman and Steel by showing that a version of Martin’s conjecture holds for all regressive functions on the hyperarithmetic degrees. A key step in our proof, which may have applications to other cases of Martin’s conjecture, consists of showing that we can always reduce to the case of a continuous function.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139962246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Preservation of natp","authors":"Jinhoo Ahn, Joonhee Kim, Hyoyoon Lee, Junguk Lee","doi":"10.1142/s0219061324500120","DOIUrl":"https://doi.org/10.1142/s0219061324500120","url":null,"abstract":"We prove several preservation theorems for NATP and furnish several examples of NATP. First, we prove preservation of NATP for the parametrization and sum of the theories of Fra\"{i}ss'{e} limits of Fra\"{i}ss'{e} classes satisfying strong amalgamation property. Second, we prove preservation of NATP for two kinds of dense/co-dense expansions, that is, the theories of lovely pairs and of H-structures for geometric theories and dense/co-dense expansion on vector spaces. Third, we prove preservation of NATP for the generic predicate expansion and the pair of an algebraically closed field and its distinguished subfield; for the latter, not only NATP, but also preservations of NTP$_1$ and NTP$_2$ are considered. Fourth, we present some proper examples of NATP using the results proved in this paper. Most of all, we show that the model companion of the theory of algebraically closed fields with circular orders (ACFO) is NATP.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Enriching a predicate and tame expansions of the integers","authors":"Gabriel Conant, Christian d'Elbée, Yatir Halevi, Léo Jimenez, Silvain Rideau-Kikuchi","doi":"10.1142/s0219061324500119","DOIUrl":"https://doi.org/10.1142/s0219061324500119","url":null,"abstract":"Given a structure $mathcal{M}$ and a stably embedded $emptyset$-definable set $Q$, we prove tameness preservation results when enriching the induced structure on $Q$ by some further structure $mathcal{Q}$. In particular, we show that if $T=text{Th}(mathcal{M})$ and $text{Th}(mathcal{Q})$ are stable (resp., superstable, $omega$-stable), then so is the theory $T[mathcal{Q}]$ of the enrichment of $mathcal{M}$ by $mathcal{Q}$. Assuming simplicity of $T$, elimination of hyperimaginaries and a further condition on $Q$ related to the behavior of algebraic closure, we also show that simplicity and NSOP$_1$ pass from $text{Th}(mathcal{Q})$ to $T[mathcal{Q}]$. We then prove several applications for tame expansions of weakly minimal structures and, in particular, the group of integers. For example, we construct the first known examples of strictly stable expansions of $(mathbb{Z},+)$. More generally, we show that any stable (resp., superstable, simple, NIP, NTP$_2$, NSOP$_1$) countable graph can be defined in a stable (resp., superstable, simple, NIP, NTP$_2$, NSOP$_1$) expansion of $(mathbb{Z},+)$ by some unary predicate $Asubseteqmathbb{N}$.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}