Pub Date : 2019-05-21DOI: 10.1142/S0219061321500136
B. Monin, Ludovic Patey
The infinite pigeonhole principle for 2-partitions ([Formula: see text]) asserts the existence, for every set [Formula: see text], of an infinite subset of [Formula: see text] or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that [Formula: see text] admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every [Formula: see text] set, of an infinite low[Formula: see text] subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak et al.
{"title":"The weakness of the pigeonhole principle under hyperarithmetical reductions","authors":"B. Monin, Ludovic Patey","doi":"10.1142/S0219061321500136","DOIUrl":"https://doi.org/10.1142/S0219061321500136","url":null,"abstract":"The infinite pigeonhole principle for 2-partitions ([Formula: see text]) asserts the existence, for every set [Formula: see text], of an infinite subset of [Formula: see text] or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that [Formula: see text] admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every [Formula: see text] set, of an infinite low[Formula: see text] subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak et al.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78839612","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}
Pub Date : 2019-05-20DOI: 10.1142/s0219061320500166
Daniel Turetsky
Using new techniques for controlling the categoricity spectrum of a structure, we construct a structure with degree of categoricity but infinite spectral dimension, answering a question of Bazhenov, Kalimullin and Yamaleev. Using the same techniques, we construct a computably categorical structure of non-computable Scott rank.
{"title":"Coding in the automorphism group of a computably categorical structure","authors":"Daniel Turetsky","doi":"10.1142/s0219061320500166","DOIUrl":"https://doi.org/10.1142/s0219061320500166","url":null,"abstract":"Using new techniques for controlling the categoricity spectrum of a structure, we construct a structure with degree of categoricity but infinite spectral dimension, answering a question of Bazhenov, Kalimullin and Yamaleev. Using the same techniques, we construct a computably categorical structure of non-computable Scott rank.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86390027","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}
Pub Date : 2019-05-17DOI: 10.1142/s0219061321500070
A. Fornasiero, E. Kaplan
Let $T$ be a complete, model complete o-minimal theory extending the theory RCF of real closed ordered fields in some appropriate language $L$. We study derivations $delta$ on models $mathcal{M}models T$. We introduce the notion of a $T$-derivation: a derivation which is compatible with the $L(emptyset)$-definable $mathcal{C}^1$-functions on $mathcal{M}$. We show that the theory of $T$-models with a $T$-derivation has a model completion $T^delta_G$. The derivation in models $(mathcal{M},delta)models T^delta_G$ behaves "generically," it is wildly discontinuous and its kernel is a dense elementary $L$-substructure of $mathcal{M}$. If $T =$ RCF, then $T^delta_G$ is the theory of closed ordered differential fields (CODF) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that $T^delta_G$ has $T$ as its open core, that $T^delta_G$ is distal, and that $T^delta_G$ eliminates imaginaries. We also show that the theory of $T$-models with finitely many commuting $T$-derivations has a model completion.
{"title":"Generic derivations on o-minimal structures","authors":"A. Fornasiero, E. Kaplan","doi":"10.1142/s0219061321500070","DOIUrl":"https://doi.org/10.1142/s0219061321500070","url":null,"abstract":"Let $T$ be a complete, model complete o-minimal theory extending the theory RCF of real closed ordered fields in some appropriate language $L$. We study derivations $delta$ on models $mathcal{M}models T$. We introduce the notion of a $T$-derivation: a derivation which is compatible with the $L(emptyset)$-definable $mathcal{C}^1$-functions on $mathcal{M}$. We show that the theory of $T$-models with a $T$-derivation has a model completion $T^delta_G$. The derivation in models $(mathcal{M},delta)models T^delta_G$ behaves \"generically,\" it is wildly discontinuous and its kernel is a dense elementary $L$-substructure of $mathcal{M}$. If $T =$ RCF, then $T^delta_G$ is the theory of closed ordered differential fields (CODF) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that $T^delta_G$ has $T$ as its open core, that $T^delta_G$ is distal, and that $T^delta_G$ eliminates imaginaries. We also show that the theory of $T$-models with finitely many commuting $T$-derivations has a model completion.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76135318","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}
Pub Date : 2019-04-23DOI: 10.1142/s021906132150015x
Shlomo Eshel, Itay Kaplan
Combining two results from machine learning theory we prove that a formula is NIP if and only if it satisfies uniform definability of types over finite sets (UDTFS). This settles a conjecture of Laskowski.
{"title":"On uniform definability of types over finite sets for NIP formulas","authors":"Shlomo Eshel, Itay Kaplan","doi":"10.1142/s021906132150015x","DOIUrl":"https://doi.org/10.1142/s021906132150015x","url":null,"abstract":"Combining two results from machine learning theory we prove that a formula is NIP if and only if it satisfies uniform definability of types over finite sets (UDTFS). This settles a conjecture of Laskowski.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87288760","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}
Pub Date : 2019-04-23DOI: 10.1142/S0219061321500276
Thomas Gilton, I. Neeman
Author(s): Gilton, Thomas; Neeman, Itay | Abstract: We show that the Abraham-Rubin-Shelah Open Coloring Axiom is consistent with a large continuum, in particular, consistent with $2^{aleph_0}=aleph_3$. This answers one of the main open questions from the 1985 paper of Abraham-Rubin-Shelah. As in their paper, we need to construct names for so-called preassignments of colors in order to add the necessary homogeneous sets. However, these names are constructed over models satisfying the CH. In order to address this difficulty, we show how to construct such names with very strong symmetry conditions. This symmetry allows us to combine them in many different ways, using a new type of poset called a Partition Product, and thereby obtain a model of this axiom in which $2^{aleph_0}=aleph_3$.
{"title":"Abraham-Rubin-Shelah open colorings and a large continuum","authors":"Thomas Gilton, I. Neeman","doi":"10.1142/S0219061321500276","DOIUrl":"https://doi.org/10.1142/S0219061321500276","url":null,"abstract":"Author(s): Gilton, Thomas; Neeman, Itay | Abstract: We show that the Abraham-Rubin-Shelah Open Coloring Axiom is consistent with a large continuum, in particular, consistent with $2^{aleph_0}=aleph_3$. This answers one of the main open questions from the 1985 paper of Abraham-Rubin-Shelah. As in their paper, we need to construct names for so-called preassignments of colors in order to add the necessary homogeneous sets. However, these names are constructed over models satisfying the CH. In order to address this difficulty, we show how to construct such names with very strong symmetry conditions. This symmetry allows us to combine them in many different ways, using a new type of poset called a Partition Product, and thereby obtain a model of this axiom in which $2^{aleph_0}=aleph_3$.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82272482","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}
Pub Date : 2019-04-04DOI: 10.1142/S0219061321500185
M. Goldstern, Jakob Kellner, D. Mej'ia, S. Shelah
We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new [Formula: see text]-sequences (for some regular [Formula: see text]). As an application, we show that consistently the following cardinal characteristics can be different: The (“independent”) characteristics in Cichoń’s diagram, plus [Formula: see text]. (So we get thirteen different values, including [Formula: see text] and continuum). We also give constructions to alternatively separate other MA-numbers (instead of [Formula: see text]), namely: MA for [Formula: see text]-Knaster from MA for [Formula: see text]-Knaster; and MA for the union of all [Formula: see text]-Knaster forcings from MA for precaliber.
{"title":"Controlling cardinal characteristics without adding reals","authors":"M. Goldstern, Jakob Kellner, D. Mej'ia, S. Shelah","doi":"10.1142/S0219061321500185","DOIUrl":"https://doi.org/10.1142/S0219061321500185","url":null,"abstract":"We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new [Formula: see text]-sequences (for some regular [Formula: see text]). As an application, we show that consistently the following cardinal characteristics can be different: The (“independent”) characteristics in Cichoń’s diagram, plus [Formula: see text]. (So we get thirteen different values, including [Formula: see text] and continuum). We also give constructions to alternatively separate other MA-numbers (instead of [Formula: see text]), namely: MA for [Formula: see text]-Knaster from MA for [Formula: see text]-Knaster; and MA for the union of all [Formula: see text]-Knaster forcings from MA for precaliber.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79395227","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}
Pub Date : 2019-03-20DOI: 10.1142/s021906132150029x
A. Nies, Philipp Schlicht, K. Tent
Let [Formula: see text] denote the topological group of permutations of the natural numbers. A closed subgroup [Formula: see text] of [Formula: see text] is called oligomorphic if for each [Formula: see text], its natural action on [Formula: see text]-tuples of natural numbers has only finitely many orbits. We study the complexity of the topological isomorphism relation on the oligomorphic subgroups of [Formula: see text] in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup [Formula: see text] of [Formula: see text], the coarse group [Formula: see text] is the structure with domain the cosets of open subgroups of [Formula: see text], and a ternary relation [Formula: see text]. This structure derived from [Formula: see text] was introduced in [A. Kechris, A. Nies and K. Tent, The complexity of topological group isomorphism, J. Symbolic Logic 83(3) (2018) 1190–1203, Sec. 3.3]. If [Formula: see text] has only countably many open subgroups, then [Formula: see text] is a countable structure. Coarse groups form our main tool in studying such closed subgroups of [Formula: see text]. We axiomatize them abstractly as structures with a ternary relation. For the oligomorphic groups, and also the profinite groups, we set up a Stone-type duality between the groups and the corresponding coarse groups. In particular, we can recover an isomorphic copy of [Formula: see text] from its coarse group in a Borel fashion. We use this duality to show that the isomorphism relation for oligomorphic subgroups of [Formula: see text] is Borel reducible to a Borel equivalence relation with all classes countable. We show that the same upper bound applies to the larger class of closed subgroups of [Formula: see text] that are topologically isomorphic to oligomorphic groups.
令[公式:见文]表示自然数排列的拓扑群。如果对于每个[公式:见文]，其对[公式:见文]-自然数元组的自然作用只有有限多个轨道，则[公式:见文]的封闭子群[公式:见文]被称为寡胚。在波兰空间上等价关系之间Borel约化的情况下，研究了[公式:见文]的低纯子群上拓扑同构关系的复杂性。给定[公式:见文]的一个闭子群[公式:见文]，粗群[公式:见文]是具有[公式:见文]的开子群的余集域和三元关系[公式:见文]的结构。这个结构来源于[公式:见文]，在[A]中被引入。Kechris, A. Nies和K. Tent，拓扑群同构的复杂性[j].符号逻辑83(3)(2018):190 - 1203,Sec. 3.3。如果[Formula: see text]只有可数的开放子群，则[Formula: see text]是一个可数结构。粗群是我们研究此类封闭子群的主要工具[公式:见原文]。我们将它们抽象地公理化为具有三元关系的结构。对于低纯群和无限群，我们在群和相应的粗群之间建立了stone型对偶。特别地，我们可以用Borel的方式从[Formula: see text]的粗群中恢复一个同构副本。我们利用这个对偶证明了[公式:见文]的低纯子群的同构关系是Borel可约为所有类可数的Borel等价关系。我们证明了相同的上界适用于[公式:见文本]中拓扑同构于寡纯群的更大的闭子群类。
{"title":"Coarse groups, and the isomorphism problem for oligomorphic groups","authors":"A. Nies, Philipp Schlicht, K. Tent","doi":"10.1142/s021906132150029x","DOIUrl":"https://doi.org/10.1142/s021906132150029x","url":null,"abstract":"Let [Formula: see text] denote the topological group of permutations of the natural numbers. A closed subgroup [Formula: see text] of [Formula: see text] is called oligomorphic if for each [Formula: see text], its natural action on [Formula: see text]-tuples of natural numbers has only finitely many orbits. We study the complexity of the topological isomorphism relation on the oligomorphic subgroups of [Formula: see text] in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup [Formula: see text] of [Formula: see text], the coarse group [Formula: see text] is the structure with domain the cosets of open subgroups of [Formula: see text], and a ternary relation [Formula: see text]. This structure derived from [Formula: see text] was introduced in [A. Kechris, A. Nies and K. Tent, The complexity of topological group isomorphism, J. Symbolic Logic 83(3) (2018) 1190–1203, Sec. 3.3]. If [Formula: see text] has only countably many open subgroups, then [Formula: see text] is a countable structure. Coarse groups form our main tool in studying such closed subgroups of [Formula: see text]. We axiomatize them abstractly as structures with a ternary relation. For the oligomorphic groups, and also the profinite groups, we set up a Stone-type duality between the groups and the corresponding coarse groups. In particular, we can recover an isomorphic copy of [Formula: see text] from its coarse group in a Borel fashion. We use this duality to show that the isomorphism relation for oligomorphic subgroups of [Formula: see text] is Borel reducible to a Borel equivalence relation with all classes countable. We show that the same upper bound applies to the larger class of closed subgroups of [Formula: see text] that are topologically isomorphic to oligomorphic groups.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84860043","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}
Pub Date : 2019-03-18DOI: 10.1142/S0219061320500014
F. Calderoni, H. Mildenberger, L. Ros
Answering some of the main questions from [L. Motto Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic 164(12) (2013) 1454–1492], we show that whenever [Formula: see text] is a cardinal satisfying [Formula: see text], then the embeddability relation between [Formula: see text]-sized structures is strongly invariantly universal, and hence complete for ([Formula: see text]-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or groups. This fully generalizes to the uncountable case the main results of [A. Louveau and C. Rosendal, Complete analytic equivalence relations, Trans. Amer. Math. Soc. 357(12) (2005) 4839–4866; S.-D. Friedman and L. Motto Ros, Analytic equivalence relations and bi-embeddability, J. Symbolic Logic 76(1) (2011) 243–266; J. Williams, Universal countable Borel quasi-orders, J. Symbolic Logic 79(3) (2014) 928–954; F. Calderoni and L. Motto Ros, Universality of group embeddability, Proc. Amer. Math. Soc. 146 (2018) 1765–1780].
{"title":"Uncountable structures are not classifiable up to bi-embeddability","authors":"F. Calderoni, H. Mildenberger, L. Ros","doi":"10.1142/S0219061320500014","DOIUrl":"https://doi.org/10.1142/S0219061320500014","url":null,"abstract":"Answering some of the main questions from [L. Motto Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic 164(12) (2013) 1454–1492], we show that whenever [Formula: see text] is a cardinal satisfying [Formula: see text], then the embeddability relation between [Formula: see text]-sized structures is strongly invariantly universal, and hence complete for ([Formula: see text]-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or groups. This fully generalizes to the uncountable case the main results of [A. Louveau and C. Rosendal, Complete analytic equivalence relations, Trans. Amer. Math. Soc. 357(12) (2005) 4839–4866; S.-D. Friedman and L. Motto Ros, Analytic equivalence relations and bi-embeddability, J. Symbolic Logic 76(1) (2011) 243–266; J. Williams, Universal countable Borel quasi-orders, J. Symbolic Logic 79(3) (2014) 928–954; F. Calderoni and L. Motto Ros, Universality of group embeddability, Proc. Amer. Math. Soc. 146 (2018) 1765–1780].","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79202330","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}
Pub Date : 2019-02-26DOI: 10.1142/s0219061320500245
Will Boney, M. Lieberman
We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also equivalent to various forms of tameness for abstract elementary classes. This systematizes and extends results of [W. Boney and S. Unger, Large cardinal axioms from tameness in AECs, Proc. Amer. Math. Soc. 145(10) (2017) 4517–4532; A. Brooke-Taylor and J. Rosický, Accessible images revisited, Proc. AMS 145(3) (2016) 1317–1327; M. Lieberman, A category-theoretic characterization of almost measurable cardinals (Submitted, 2018), http://arxiv.org/abs/1809.06963; M. Lieberman and J. Rosický, Classification theory for accessible categories. J. Symbolic Logic 81(1) (2016) 1647–1648].
{"title":"Tameness, powerful images, and large cardinals","authors":"Will Boney, M. Lieberman","doi":"10.1142/s0219061320500245","DOIUrl":"https://doi.org/10.1142/s0219061320500245","url":null,"abstract":"We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also equivalent to various forms of tameness for abstract elementary classes. This systematizes and extends results of [W. Boney and S. Unger, Large cardinal axioms from tameness in AECs, Proc. Amer. Math. Soc. 145(10) (2017) 4517–4532; A. Brooke-Taylor and J. Rosický, Accessible images revisited, Proc. AMS 145(3) (2016) 1317–1327; M. Lieberman, A category-theoretic characterization of almost measurable cardinals (Submitted, 2018), http://arxiv.org/abs/1809.06963; M. Lieberman and J. Rosický, Classification theory for accessible categories. J. Symbolic Logic 81(1) (2016) 1647–1648].","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81536829","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}
Pub Date : 2019-02-15DOI: 10.1142/S0219061319500132
Sandra Müller, R. Schindler, W. Woodin
We prove the following result which is due to the third author. Let [Formula: see text]. If [Formula: see text] determinacy and [Formula: see text] determinacy both hold true and there is no [Formula: see text]-definable [Formula: see text]-sequence of pairwise distinct reals, then [Formula: see text] exists and is [Formula: see text]-iterable. The proof yields that [Formula: see text] determinacy implies that [Formula: see text] exists and is [Formula: see text]-iterable for all reals [Formula: see text]. A consequence is the Determinacy Transfer Theorem for arbitrary [Formula: see text], namely the statement that [Formula: see text] determinacy implies [Formula: see text] determinacy.
{"title":"Mice with finitely many Woodin cardinals from optimal determinacy hypotheses","authors":"Sandra Müller, R. Schindler, W. Woodin","doi":"10.1142/S0219061319500132","DOIUrl":"https://doi.org/10.1142/S0219061319500132","url":null,"abstract":"We prove the following result which is due to the third author. Let [Formula: see text]. If [Formula: see text] determinacy and [Formula: see text] determinacy both hold true and there is no [Formula: see text]-definable [Formula: see text]-sequence of pairwise distinct reals, then [Formula: see text] exists and is [Formula: see text]-iterable. The proof yields that [Formula: see text] determinacy implies that [Formula: see text] exists and is [Formula: see text]-iterable for all reals [Formula: see text]. A consequence is the Determinacy Transfer Theorem for arbitrary [Formula: see text], namely the statement that [Formula: see text] determinacy implies [Formula: see text] determinacy.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2019-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83237675","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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