Authors:Mohammad Golshani Pages: 3 - 25 Abstract: Assume \(\lambda \) is a singular limit of \(\eta \) supercompact cardinals, where \(\eta \le \lambda \) is a limit ordinal. We present two methods for arranging the tree property to hold at \(\lambda ^{+}\) while making \(\lambda ^{+}\) the successor of the limit of the first \(\eta \) measurable cardinals. The first method is then used to get, from the same assumptions, the tree property at \(\aleph _{\eta ^2+1}\) with the failure of SCH at \(\aleph _{\eta ^2}\) . This extends results of Neeman and Sinapova. The second method is also used to get the tree property at the successor of an arbitrary singular cardinal, which extends some results of Magidor–Shelah, Neeman and Sinapova. PubDate: 2018-02-01 DOI: 10.1007/s00153-017-0581-4 Issue No:Vol. 57, No. 1-2 (2018)

Authors:Mohammad Golshani; Shahram Mohsenipour Pages: 27 - 35 Abstract: Assuming the existence of a Mahlo cardinal, we produce a generic extension of Gödel’s constructible universe L, in which the \(\textit{GCH}\) holds and the transfer principles \((\aleph _2, \aleph _0) \rightarrow (\aleph _3, \aleph _1)\) and \((\aleph _3, \aleph _1) \rightarrow (\aleph _2, \aleph _0)\) fail simultaneously. The result answers a question of Silver from 1971. We also extend our result to higher gaps. PubDate: 2018-02-01 DOI: 10.1007/s00153-017-0586-z Issue No:Vol. 57, No. 1-2 (2018)

Authors:Alireza Mofidi Pages: 37 - 71 Abstract: We investigate some dynamical features of the actions of automorphisms in the context of model theory. We interpret a few notions such as compact systems, entropy and symbolic representations from the theory of dynamical systems in the realm of model theory. In this direction, we settle a number of characterizations of NIP theories in terms of dynamics of automorphisms and invariant measures. For example, it is shown that the property of NIP corresponds to the compactness property of some associated systems and also to the zero entropy property of automorphisms. These results give a correspondence between some notions of tameness in model theory and ergodic theory. Moreover, we study the concept of symbolic representation and consider it in some well known mathematical objects such as the circle group, Bohr sets, Sturmian sequences, the structure \((\mathbb {Z},+,U)\) , and random graphs with a model theoretic point of view in mind. We establish certain characterizations for stability theoretic dividing lines, such as independence property, order property and strict order property in terms of associated symbolic representations. At the end, we propose some applications of symbolic representations and these characterizations by giving a proof for a classical theorem by Shelah and also introducing some invariants associated to the types and elements of models. PubDate: 2018-02-01 DOI: 10.1007/s00153-017-0580-5 Issue No:Vol. 57, No. 1-2 (2018)

Authors:Ralf Schindler Pages: 73 - 82 Abstract: Generalizing Woodin’s extender algebra, cf. e.g. Steel (in: Kanamori (ed) Handbook of set theory, Springer, Berlin, 2010), we isolate the long extender algebra as a general version of Bukowský’s forcing, cf. Bukovský (Fundam Math 83:35–46, 1973), in the presence of a supercompact cardinal. PubDate: 2018-02-01 DOI: 10.1007/s00153-017-0585-0 Issue No:Vol. 57, No. 1-2 (2018)

Authors:James Cummings Pages: 83 - 90 Abstract: Monroe Eskew (Tree properties on \(\omega _1\) and \(\omega _2\) , 2016. https://mathoverflow.net/questions/217951/tree-properties-on-omega-1-and-omega-2) asked whether the tree property at \(\omega _2\) implies there is no Kurepa tree (as is the case in the Mitchell model, or under PFA). We prove that the tree property at \(\omega _2\) is consistent with the existence of \(\omega _1\) -trees with as many branches as desired. PubDate: 2018-02-01 DOI: 10.1007/s00153-017-0579-y Issue No:Vol. 57, No. 1-2 (2018)

Authors:Ali Enayat; Matt Kaufmann; Zachiri McKenzie Pages: 91 - 139 Abstract: Given a model \(\mathcal {M}\) of set theory, and a nontrivial automorphism j of \(\mathcal {M}\) , let \(\mathcal {I}_{\mathrm {fix}}(j)\) be the submodel of \(\mathcal {M}\) whose universe consists of elements m of \(\mathcal {M}\) such that \(j(x)=x\) for every x in the transitive closure of m (where the transitive closure of m is computed within \(\mathcal {M}\) ). Here we study the class \(\mathcal {C}\) of structures of the form \(\mathcal {I}_{\mathrm {fix}}(j)\) , where the ambient model \(\mathcal {M}\) satisfies a frugal yet robust fragment of \(\mathrm {ZFC}\) known as \(\mathrm {MOST}\) , and \(j(m)=m\) whenever m is a finite ordinal in the sense of \(\mathcal {M}.\) Our main achievement is the calculation of the theory of \(\mathcal {C}\) as precisely \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\) - \(\mathrm {Collection}\) . The following theorems encapsulate our principal results: Theorem A. Every structure in \(\mathcal {C}\) satisfies \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\) - \(\mathrm { Collection}\) . Theorem B. Each of the following three conditions is sufficient for a countable structure \(\mathcal {N}\) to be in \(\mathcal {C}\) : (a) \(\mathcal {N}\) is a transitive model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\) - \(\mathrm {Collection}\) . (b) \(\mathcal {N}\) is a recursively saturated model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\) - PubDate: 2018-02-01 DOI: 10.1007/s00153-017-0582-3 Issue No:Vol. 57, No. 1-2 (2018)

Authors:Zaniar Ghadernezhad; Andrés Villaveces Pages: 141 - 157 Abstract: We prove a version of a small index property theorem for strong amalgamation classes. Our result builds on an earlier theorem by Lascar and Shelah (in their case, for saturated models of uncountable first-order theories). We then study versions of the small index property for various non-elementary classes. In particular, we obtain the small index property for quasiminimal pregeometry structures. PubDate: 2018-02-01 DOI: 10.1007/s00153-017-0587-y Issue No:Vol. 57, No. 1-2 (2018)

Authors:Dugald Macpherson Pages: 159 - 184 Abstract: This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first-order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory. PubDate: 2018-02-01 DOI: 10.1007/s00153-017-0584-1 Issue No:Vol. 57, No. 1-2 (2018)

Authors:Stevo Todorcevic Pages: 185 - 194 Abstract: We examine the differences between three standard classes of forcing notions relative to the way they collapse the continuum. It turns out that proper and semi-proper posets behave differently in that respect from the class of posets that preserve stationary subsets of \(\omega _1\) . PubDate: 2018-02-01 DOI: 10.1007/s00153-017-0588-x Issue No:Vol. 57, No. 1-2 (2018)

Authors:Josef Berger; Gregor Svindland Abstract: We show constructively that every quasi-convex, uniformly continuous function \(f:C \rightarrow \mathbb {R}\) with at most one minimum point has a minimum point, where C is a convex compact subset of a finite dimensional normed space. Applications include a result on strictly quasi-convex functions, a supporting hyperplane theorem, and a short proof of the constructive fundamental theorem of approximation theory. PubDate: 2018-02-22 DOI: 10.1007/s00153-018-0619-2

Authors:Gunter Fuchs Abstract: The effects of (bounded versions of) the forcing axioms \(\mathsf {SCFA}\) , \(\mathsf {PFA}\) and \(\mathsf {MM}\) on the failure of weak threaded square principles of the form \(\square (\lambda ,\kappa )\) are analyzed. To this end, a diagonal reflection principle, \(\mathsf {DSR}{\left( {<}\kappa ,S\right) }\) is introduced. It is shown that \(\mathsf {SCFA} \) implies \(\mathsf {DSR}{\left( \omega _1,S^\lambda _\omega \right) }\) , for all regular \(\lambda \ge \omega _2\) , and that \(\mathsf {DSR}{\left( \omega _1,S^\lambda _\omega \right) }\) implies the failure of \(\square (\lambda ,\omega _1)\) if \(\lambda >\omega _2\) , and it implies the failure of \(\square (\lambda ,\omega )\) if \(\lambda =\omega _2\) . It is also shown that this result is sharp. It is noted that \(\mathsf {MM}\) / \(\mathsf {PFA}\) imply the failure of \(\square (\lambda ,\omega _1)\) , for every regular \(\lambda >\omega _1\) , and that this result is sharp as well. PubDate: 2018-02-19 DOI: 10.1007/s00153-018-0614-7

Authors:Taishi Kurahashi Abstract: Let \(\varGamma \) be a class of formulas. We say that a theory T in classical logic has the \(\varGamma \) -disjunction property if for any \(\varGamma \) sentences \(\varphi \) and \(\psi \) , either \(T \vdash \varphi \) or \(T \vdash \psi \) whenever \(T \vdash \varphi \vee \psi \) . First, we characterize the \(\varGamma \) -disjunction property in terms of the notion of partial conservativity. Secondly, we prove a model theoretic characterization result for \(\varSigma _n\) -disjunction property. Thirdly, we investigate relationships between partial disjunction properties and several other properties of theories containing Peano arithmetic. Finally, we investigate unprovability of formalized partial disjunction properties. PubDate: 2018-02-14 DOI: 10.1007/s00153-018-0618-3

Authors:Yuan Yuan Zheng Abstract: We construct a collection of new topological Ramsey spaces of trees. It is based on the Halpern-Läuchli theorem, but different from the Milliken space of strong subtrees. We give an example of its application by proving a partition theorem for profinite graphs. PubDate: 2018-02-10 DOI: 10.1007/s00153-018-0617-4

Authors:Ruggero Pagnan Abstract: By splitting idempotent morphisms in the total and base categories of fibrations we provide an explicit elementary description of the Cauchy completion of objects in the categories Fib( \(\mathbb {B}\) ) of fibrations with a fixed base category \(\mathbb {B}\) and Fib of fibrations with any base category. Two universal constructions are at issue, corresponding to two fibered reflections involving the fibration of fibrations \(\mathbf{Fib}\rightarrow \mathbf{Cat}\) . PubDate: 2018-02-06 DOI: 10.1007/s00153-018-0616-5

Authors:Maxwell Levine Abstract: Current research in set theory raises the possibility that \(\square _{\kappa ,<\lambda }\) can be made compatible with some stationary reflection, depending on the parameter \(\lambda \) . The purpose of this paper is to demonstrate the difficulty in such results. We prove that the poset \({\mathbb {S}}(\kappa ,<\lambda )\) , which adds a \(\square _{\kappa ,<\lambda }\) -sequence by initial segments, will also add non-reflecting stationary sets concentrating in any given cofinality below \(\kappa \) . We also investigate the CMB poset, which adds \(\square _\kappa ^*\) in a slightly different way. We prove that the CMB poset also adds non-reflecting stationary sets, but not necessarily concentrating in any cofinality. PubDate: 2018-02-02 DOI: 10.1007/s00153-018-0613-8

Authors:S. S. Goncharov; J. F. Knight; I. Souldatos Abstract: The Hanf number for a set S of sentences in \(\mathcal {L}_{\omega _1,\omega }\) (or some other logic) is the least infinite cardinal \(\kappa \) such that for all \(\varphi \in S\) , if \(\varphi \) has models in all infinite cardinalities less than \(\kappa \) , then it has models of all infinite cardinalities. Friedman asked what is the Hanf number for Scott sentences of computable structures. We show that the value is \(\beth _{\omega _1^{CK}}\) . The same argument proves that \(\beth _{\omega _1^{CK}}\) is the Hanf number for Scott sentences of hyperarithmetical structures. PubDate: 2018-02-01 DOI: 10.1007/s00153-018-0615-6

Authors:Jan Dobrowolski Abstract: We prove a preservation theorem for NTP \(_1\) in the context of the generic variations construction. We also prove that NTP \(_1\) is preserved under adding to a geometric theory a generic predicate. PubDate: 2018-01-27 DOI: 10.1007/s00153-018-0609-4

Authors:Hajime Ishihara; Maria Emilia Maietti; Samuele Maschio; Thomas Streicher Abstract: Consistency with the formal Church’s thesis, for short CT, and the axiom of choice, for short AC, was one of the requirements asked to be satisfied by the intensional level of a two-level foundation for constructive mathematics as proposed by Maietti and Sambin (in Crosilla, Schuster (eds) From sets and types to topology and analysis: practicable foundations for constructive mathematics, Oxford University Press, Oxford, 2005). Here we show that this is the case for the intensional level of the two-level Minimalist Foundation, for short MF, completed in 2009 by the second author. The intensional level of MF consists of an intensional type theory à la Martin-Löf, called mTT. The consistency of mTT with CT and AC is obtained by showing the consistency with the formal Church’s thesis of a fragment of intensional Martin-Löf’s type theory, called \(\mathbf{MLtt}_1\) , where mTT can be easily interpreted. Then to show the consistency of \(\mathbf{MLtt}_1\) with CT we interpret it within Feferman’s predicative theory of non-iterative fixpoints \(\widehat{ID_1}\) by extending the well known Kleene’s realizability semantics of intuitionistic arithmetics so that CT is trivially validated. More in detail the fragment \(\mathbf{MLtt}_1\) we interpret consists of first order intensional Martin-Löf’s type theory with one universe and with explicit substitution rules in place of usual equality rules preserving type constructors (hence without the so called \(\xi \) -rule which is not valid in our realizability semantics). A key difficulty encountered in our interpretation was to use the right interpretation of lambda abstraction in the applicative structure of natural numbers in order to model all the equality rules of \(\mathbf{MLtt}_1\) correctly. In particular the universe of \(\mathbf{MLtt}_1\) is modelled by means of \(\widehat{ID_1}\) -fixpoints following a technique due first to Aczel and used by Feferman and Beeson. PubDate: 2018-01-27 DOI: 10.1007/s00153-018-0612-9

Authors:Gunter Fuchs Abstract: For an ordinal \(\varepsilon \) , I introduce a variant of the notion of subcompleteness of a forcing poset, which I call \(\varepsilon \) -subcompleteness, and show that this class of forcings enjoys some closure properties that the original class of subcomplete forcings does not seem to have: factors of \(\varepsilon \) -subcomplete forcings are \(\varepsilon \) -subcomplete, and if \(\mathbb {P}\) and \(\mathbb {Q}\) are forcing-equivalent notions, then \(\mathbb {P}\) is \(\varepsilon \) -subcomplete iff \(\mathbb {Q}\) is. I formulate a Two Step Theorem for \(\varepsilon \) -subcompleteness and prove an RCS iteration theorem for \(\varepsilon \) -subcompleteness which is slightly less restrictive than the original one, in that its formulation is more careful about the amount of collapsing necessary. Finally, I show that an adequate degree of \(\varepsilon \) -subcompleteness follows from the \(\kappa \) -distributivity of a forcing, for \(\kappa >\omega _1\) . PubDate: 2018-01-17 DOI: 10.1007/s00153-018-0611-x