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Journal of Mathematical Logic最新文献

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The domination monoid in o-minimal theories o-极小理论中的支配单群
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-08-04 DOI: 10.1142/S0219061321500306
Rosario Mennuni
{"title":"The domination monoid in o-minimal theories","authors":"Rosario Mennuni","doi":"10.1142/S0219061321500306","DOIUrl":"https://doi.org/10.1142/S0219061321500306","url":null,"abstract":"We study the monoid of global invariant types modulo domination-equivalence in the context of o-minimal theories. We reduce its computation to the problem of proving that it is generated by classes of [Formula: see text]-types. We show this to hold in Real Closed Fields, where generators of this monoid correspond to invariant convex subrings of the monster model. Combined with [C. Ealy, D. Haskell and J. Maríková, Residue field domination in real closed valued fields, Notre Dame J. Formal Logic 60(3) (2019) 333–351], this allows us to compute the domination monoid in the weakly o-minimal theory of Real Closed Valued Fields.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"10 1","pages":"2150030:1-2150030:36"},"PeriodicalIF":0.9,"publicationDate":"2020-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78954872","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}
引用次数: 4
Reduction games, provability and compactness 约简对策，可证明性和紧凑性
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-08-03 DOI: 10.1142/s021906132250009x
D. Dzhafarov, D. Hirschfeldt, Sarah C. Reitzes
{"title":"Reduction games, provability and compactness","authors":"D. Dzhafarov, D. Hirschfeldt, Sarah C. Reitzes","doi":"10.1142/s021906132250009x","DOIUrl":"https://doi.org/10.1142/s021906132250009x","url":null,"abstract":"Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between [Formula: see text] principles over [Formula: see text]-models of [Formula: see text]. They also introduced a version of this game that similarly captures provability over [Formula: see text]. We generalize and extend this game-theoretic framework to other formal systems, and establish a certain compactness result that shows that if an implication [Formula: see text] between two principles holds, then there exists a winning strategy that achieves victory in a number of moves bounded by a number independent of the specific run of the game. This compactness result generalizes an old proof-theoretic fact noted by H. Wang (1981), and has applications to the reverse mathematics of combinatorial principles. We also demonstrate how this framework leads to a new kind of analysis of the logical strength of mathematical problems that refines both that of reverse mathematics and that of computability-theoretic notions such as Weihrauch reducibility, allowing for a kind of fine-structural comparison between [Formula: see text] principles that has both computability-theoretic and proof-theoretic aspects, and can help us distinguish between these, for example by showing that a certain use of a principle in a proof is “purely proof-theoretic”, as opposed to relying on its computability-theoretic strength. We give examples of this analysis to a number of principles at the level of [Formula: see text], uncovering new differences between their logical strengths.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"2 1","pages":"2250009:1-2250009:37"},"PeriodicalIF":0.9,"publicationDate":"2020-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76310105","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}
引用次数: 2
Inner models from extended logics: Part 1 扩展逻辑的内部模型:第1部分
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-07-21 DOI: 10.1142/S0219061321500124
J. Kennedy, M. Magidor, Jouko Vaananen
{"title":"Inner models from extended logics: Part 1","authors":"J. Kennedy, M. Magidor, Jouko Vaananen","doi":"10.1142/S0219061321500124","DOIUrl":"https://doi.org/10.1142/S0219061321500124","url":null,"abstract":"We introduce a new inner model $C(aa)$ arising from stationary logic. We show that assuming a proper class of Woodin cardinals, or alternatively $MM^{++}$, the regular uncountable cardinals of $V$ are measurable in the inner model $C(aa)$, the theory of $C(aa)$ is (set) forcing absolute, and $C(aa)$ satisfies CH. We introduce an auxiliary concept that we call club determinacy, which simplifies the construction of $C(aa)$ greatly but may have also independent interest. Based on club determinacy, we introduce the concept of aa-mouse which we use to prove CH and other properties of the inner model $C(aa)$.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"63 1","pages":"2150012:1-2150012:53"},"PeriodicalIF":0.9,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74793119","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}
引用次数: 5
Definable completeness of P-minimal fields and applications p -极小域的可定义完备性及其应用
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-07-15 DOI: 10.1142/s0219061322500040
Pablo Cubides Kovacsics, Françoise Delon
{"title":"Definable completeness of P-minimal fields and applications","authors":"Pablo Cubides Kovacsics, Françoise Delon","doi":"10.1142/s0219061322500040","DOIUrl":"https://doi.org/10.1142/s0219061322500040","url":null,"abstract":"We show that every definable nested family of closed and bounded subsets of a [Formula: see text]-minimal field [Formula: see text] has nonempty intersection. As an application we answer a question of Darnière and Halupczok showing that [Formula: see text]-minimal fields satisfy the “extreme value property”: for every closed and bounded subset [Formula: see text] and every interpretable continuous function [Formula: see text] (where [Formula: see text] denotes the value group), [Formula: see text] admits a maximal value. Two further corollaries are obtained as a consequence of their work. The first one shows that every interpretable subset of [Formula: see text] is already interpretable in the language of rings, answering a question of Cluckers and Halupczok. This implies in particular that every [Formula: see text]-minimal field is polynomially bounded. The second one characterizes those [Formula: see text]-minimal fields satisfying a classical cell preparation theorem as those having definable Skolem functions, generalizing a result of Mourgues.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"65 1","pages":"2250004:1-2250004:16"},"PeriodicalIF":0.9,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86372251","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}
引用次数: 0
Ramsey's theorem for pairs, collection, and proof size 拉姆齐定理对，集合，和证明大小
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-05-14 DOI: 10.1142/s0219061323500071
L. Kolodziejczyk, Tin Lok Wong, K. Yokoyama
{"title":"Ramsey's theorem for pairs, collection, and proof size","authors":"L. Kolodziejczyk, Tin Lok Wong, K. Yokoyama","doi":"10.1142/s0219061323500071","DOIUrl":"https://doi.org/10.1142/s0219061323500071","url":null,"abstract":"We prove that any proof of a $forall Sigma^0_2$ sentence in the theory $mathrm{WKL}_0 + mathrm{RT}^2_2$ can be translated into a proof in $mathrm{RCA}_0$ at the cost of a polynomial increase in size. In fact, the proof in $mathrm{RCA}_0$ can be found by a polynomial-time algorithm. On the other hand, $mathrm{RT}^2_2$ has non-elementary speedup over the weaker base theory $mathrm{RCA}^*_0$ for proofs of $Sigma_1$ sentences. \u0000We also show that for $n ge 0$, proofs of $Pi_{n+2}$ sentences in $mathrm{B}Sigma_{n+1}+exp$ can be translated into proofs in $mathrm{I}Sigma_{n} + exp$ at polynomial cost. Moreover, the $Pi_{n+2}$-conservativity of $mathrm{B}Sigma_{n+1} + exp$ over $mathrm{I}Sigma_{n} + exp$ can be proved in $mathrm{PV}$, a fragment of bounded arithmetic corresponding to polynomial-time computation. For $n ge 1$, this answers a question of Clote, Hajek, and Paris.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43520843","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}
引用次数: 3
The consistency strength of hyperstationarity 超平稳的一致性强度
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-04-01 DOI: 10.1142/S021906132050004X
J. Bagaria, M. Magidor, Salvador Mancilla
{"title":"The consistency strength of hyperstationarity","authors":"J. Bagaria, M. Magidor, Salvador Mancilla","doi":"10.1142/S021906132050004X","DOIUrl":"https://doi.org/10.1142/S021906132050004X","url":null,"abstract":"We introduce the large-cardinal notions of [Formula: see text]-greatly-Mahlo and [Formula: see text]-reflection cardinals and prove (1) in the constructible universe, [Formula: see text], the first [Formula: see text]-reflection cardinal, for [Formula: see text] a successor ordinal, is strictly between the first [Formula: see text]-greatly-Mahlo and the first [Formula: see text]-indescribable cardinals, (2) assuming the existence of a [Formula: see text]-reflection cardinal [Formula: see text] in [Formula: see text], [Formula: see text] a successor ordinal, there exists a forcing notion in [Formula: see text] that preserves cardinals and forces that [Formula: see text] is [Formula: see text]-stationary, which implies that the consistency strength of the existence of a [Formula: see text]-stationary cardinal is strictly below a [Formula: see text]-indescribable cardinal. These results generalize to all successor ordinals [Formula: see text] the original same result of Mekler–Shelah [A. Mekler and S. Shelah, The consistency strength of every stationary set reflects, Israel J. Math. 67(3) (1989) 353–365] about a [Formula: see text]-stationary cardinal, i.e. a cardinal that reflects all its stationary sets.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"55 1","pages":"2050004"},"PeriodicalIF":0.9,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76158393","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}
引用次数: 3
Equivalence relations and determinacy 等价关系和确定性
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-03-04 DOI: 10.1142/s0219061322500039
Logan Crone, Lior Fishman, Stephen Jackson
{"title":"Equivalence relations and determinacy","authors":"Logan Crone, Lior Fishman, Stephen Jackson","doi":"10.1142/s0219061322500039","DOIUrl":"https://doi.org/10.1142/s0219061322500039","url":null,"abstract":"We introduce the notion of $(Gamma,E)$-determinacy for $Gamma$ a pointclass and $E$ an equivalence relation on a Polish space $X$. A case of particular interest is the case when $E=E_G$ is the (left) shift-action of $G$ on $S^G$ where $S=2={0,1}$ or $S=omega$. We show that for all shift actions by countable groups $G$, and any \"reasonable\" pointclass $Gamma$, that $(Gamma,E_G)$-determinacy implies $Gamma$-determinacy. We also prove a corresponding result when $E$ is a subshift of finite type of the shift map on $2^mathbb{Z}$.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"6 1","pages":"2250003:1-2250003:19"},"PeriodicalIF":0.9,"publicationDate":"2020-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86482900","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}
引用次数: 1
On the existence of small antichains for definable quasi-orders 关于可定义拟序的小反链的存在性
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-02-15 DOI: 10.1142/s0219061321500057
Raphaël Carroy, B. D. Miller, Zolt'an Vidny'anszky
{"title":"On the existence of small antichains for definable quasi-orders","authors":"Raphaël Carroy, B. D. Miller, Zolt'an Vidny'anszky","doi":"10.1142/s0219061321500057","DOIUrl":"https://doi.org/10.1142/s0219061321500057","url":null,"abstract":"We generalize Kada’s definable strengthening of Dilworth’s characterization of the class of quasi-orders admitting an antichain of a given finite cardinality.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"8 1","pages":"2150005:1-2150005:10"},"PeriodicalIF":0.9,"publicationDate":"2020-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82342216","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}
引用次数: 1
Forking and dividing in fields with several orderings and valuations 在具有多个排序和赋值的域中进行分叉和划分
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2020-01-08 DOI: 10.1142/S0219061321500252
Will Johnson
{"title":"Forking and dividing in fields with several orderings and valuations","authors":"Will Johnson","doi":"10.1142/S0219061321500252","DOIUrl":"https://doi.org/10.1142/S0219061321500252","url":null,"abstract":"We consider existentially closed fields with several orderings, valuations, and [Formula: see text]-valuations. We show that these structures are NTP2 of finite burden, but usually have the independence property. Moreover, forking agrees with dividing, and forking can be characterized in terms of forking in ACVF, RCF, and [Formula: see text]CF.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"24 1","pages":"2150025:1-2150025:43"},"PeriodicalIF":0.9,"publicationDate":"2020-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74963473","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}
引用次数: 3
Logical laws for short existential monadic second-order sentences about graphs 关于图的存在一元二阶短句的逻辑规律
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2019-12-12 DOI: 10.1142/s0219061320500075
M. Zhukovskii
{"title":"Logical laws for short existential monadic second-order sentences about graphs","authors":"M. Zhukovskii","doi":"10.1142/s0219061320500075","DOIUrl":"https://doi.org/10.1142/s0219061320500075","url":null,"abstract":"In 2001, Le Bars proved that there exists an existential monadic second-order (EMSO) sentence such that the probability that it is true on [Formula: see text] does not converge and conjectured that, for EMSO sentences with two first-order variables, the zero–one law holds. In this paper, we prove that the conjecture fails for [Formula: see text], and give new examples of sentences with fewer variables without convergence (even for [Formula: see text]).","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"22 1","pages":"2050007:1-2050007:23"},"PeriodicalIF":0.9,"publicationDate":"2019-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72525938","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}
引用次数: 8
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