{"title":"Exact saturation in pseudo-elementary classes for simple and stable theories","authors":"Itay Kaplan, N. Ramsey, S. Shelah","doi":"10.1142/s0219061322500209","DOIUrl":"https://doi.org/10.1142/s0219061322500209","url":null,"abstract":"We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals, satisfying mild set-theoretic hypotheses, which had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analogue of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77126301","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":"Canonical fragments of the strong reflection principle","authors":"G. Fuchs","doi":"10.1142/S0219061321500239","DOIUrl":"https://doi.org/10.1142/S0219061321500239","url":null,"abstract":"For an arbitrary forcing class [Formula: see text], the [Formula: see text]-fragment of Todorčević’s strong reflection principle SRP is isolated in such a way that (1) the forcing axiom for [Formula: see text] implies the [Formula: see text]-fragment of SRP , (2) the stationary set preserving fragment of SRP is the full principle SRP , and (3) the subcomplete fragment of SRP implies the major consequences of the subcomplete forcing axiom. This fragment of SRP is consistent with CH , and even with Jensen’s principle [Formula: see text]. Along the way, some hitherto unknown effects of (the subcomplete fragment of) SRP on mutual stationarity are explored, and some limitations to the extent to which fragments of SRP may capture the effects of their corresponding forcing axioms are established.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78247006","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":"New jump operators on equivalence relations","authors":"J. Clemens, Samuel Coskey","doi":"10.1142/S0219061322500155","DOIUrl":"https://doi.org/10.1142/S0219061322500155","url":null,"abstract":"We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group $Gamma$ we introduce the $Gamma$-jump. We study the elementary properties of the $Gamma$-jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups $Gamma$, the $Gamma$-jump is emph{proper} in the sense that for any Borel equivalence relation $E$ the $Gamma$-jump of $E$ is strictly higher than $E$ in the Borel reducibility hierarchy. On the other hand there are examples of groups $Gamma$ for which the $Gamma$-jump is not proper. To establish properness, we produce an analysis of Borel equivalence relations induced by continuous actions of the automorphism group of what we denote the infinite $Gamma$-tree, and relate these to iterates of the $Gamma$-jump. We also produce several new examples of equivalence relations that arise from applying the $Gamma$-jump to classically studied equivalence relations and derive generic ergodicity results related to these. We apply our results to show that the complexity of the isomorphism problem for countable scattered linear orders properly increases with the rank.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77918601","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 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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
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