The Experts below are selected from a list of 951 Experts worldwide ranked by ideXlab platform
Saharon Shelah - One of the best experts on this subject based on the ideXlab platform.
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Double weakness
Acta Mathematica Hungarica, 2021Co-Authors: S. Garti, Saharon ShelahAbstract:We prove that, consistently, there exists a weakly but not strongly Inaccessible Cardinal $$\lambda$$ λ for which the sequence $$\langle 2^\theta:\theta
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On the non-existence of mad families
Archive for Mathematical Logic, 2019Co-Authors: Haim Horowitz, Saharon ShelahAbstract:We show that the non-existence of mad families is equiconsistent with $$\textit{ZFC}$$ ZFC , answering an old question of Mathias. We also consider the above result in the general context of maximal independent sets in Borel graphs, and we construct a Borel graph G such that $$\textit{ZF}+\textit{DC}+$$ ZF + DC + “there is no maximal independent set in G ” is equiconsistent with $$\textit{ZFC}+$$ ZFC + “there exists an Inaccessible Cardinal”.
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Examples in dependent theories
The Journal of Symbolic Logic, 2014Co-Authors: Itay Kaplan, Saharon ShelahAbstract:AbstractIn the first part we show a counterexample to a conjecture by Shelah regarding the existence of indiscernible sequences in dependent theories (up to the first Inaccessible Cardinal). In the second part we discuss generic pairs, and give an example where the pair is not dependent. Then we define the notion of directionality which deals with counting the number of coheirs of a type and we give examples of the different possibilities. Then we discuss nonsplintering, an interesting notion that appears in the work of Rami Grossberg, Andrés Villaveces and Monica VanDieren, and we show that it is not trivial (in the sense that it can be different than splitting) whenever the directionality of the theory is not small. In the appendix we study dense types in RCF.
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On CON(${\mathfrak d}_\lambda >$ cov$_\lambda$(meagre))
arXiv: Logic, 2013Co-Authors: Saharon ShelahAbstract:We prove the consistency of: for suitable strongly Inaccessible Cardinal lambda the dominating number, i.e., the cofinality of ^{lambda}lambda, is strictly bigger than cov_lambda(meagre), i.e. the minimal number of nowhere dense subsets of ^{lambda}2 needed to cover it. This answers a question of Matet.
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Examples in dependent theories
arXiv: Logic, 2010Co-Authors: Itay Kaplan, Saharon ShelahAbstract:In the first part we show a counterexample to a conjecture by Shelah regarding the existence of indiscernible sequences in dependent theories (up to the first Inaccessible Cardinal). In the second part we discuss generic pairs, and give an example where the pair is not dependent. Then we define the notion of directionality which deals with counting the number of coheirs of a type and we give examples of the different possibilities. Then we discuss non-splintering, an interesting notion that appears in the work of Rami Grossberg, Andr\'es Villaveces and Monica VanDieren, and we show that it is not trivial (in the sense that it can be different than splitting) whenever the directionality of the theory is not small. In the appendix we study dense types in RCF.
Laura Fontanella - One of the best experts on this subject based on the ideXlab platform.
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THE STRONG TREE PROPERTY AT SUCCESSORS OF SINGULAR CardinalS
The Journal of Symbolic Logic, 2014Co-Authors: Laura FontanellaAbstract:Abstract An Inaccessible Cardinal is strongly compact if, and only if, it satisfies the strong tree property. We prove that if there is a model of ZFC with infinitely many supercompact Cardinals, then there is a model of ZFC where ${\aleph _{\omega + 1}}$ has the strong tree property. Moreover, we prove that every successor of a singular limit of strongly compact Cardinals has the strong tree property.
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Strong tree properties for small Cardinals
The Journal of Symbolic Logic, 2013Co-Authors: Laura FontanellaAbstract:AbstractAn Inaccessible Cardinal κ is supercompact when (κ, λ)-ITP holds for all λ ≥ κ. We prove that if there is a model of ZFC with infinitely many supercompact Cardinals, then there is a model of ZFC where for every n ≥ 2 and μ ≥ ℕn, we have (ℕn, μ)-ITP.
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Strong tree properties for two successive Cardinals
Archive for Mathematical Logic, 2012Co-Authors: Laura FontanellaAbstract:An Inaccessible Cardinal ? is supercompact when (?, ?)-ITP holds for all ? ? ?. We prove that if there is a model of ZFC with two supercompact Cardinals, then there is a model of ZFC where simultaneously $${(\aleph_2, \mu)}$$ -ITP and $${(\aleph_3, \mu')}$$ -ITP hold, for all $${\mu\geq \aleph_2}$$ and $${\mu'\geq \aleph_3}$$ .
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Strong Tree Properties for Small Cardinals
arXiv: Logic, 2012Co-Authors: Laura FontanellaAbstract:An Inaccessible Cardinal kappa is supercompact when (kappa, lambda)-ITP holds for all lambda greater than or equal to kappa. We prove that if there is a model of ZFC with infinitely many supercompact Cardinals, then there is a model of ZFC where for every natural number n greater than 1 and for every ordinal mu greater than or equal to aleph_n, we have (aleph_n, mu)-ITP.
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Strong tree Properties for two successive Cardinals
arXiv: Logic, 2011Co-Authors: Laura FontanellaAbstract:An Inaccessible Cardinal $\kappa$ is supercompact when $(\kappa, \lambda)$-ITP holds for all $\lambda\geq \kappa.$ We prove that if there is a model of $\ZFC$ with two supercompact Cardinals, then there is a model of \ZFC where simultaneously $(\aleph_2, \mu)$-ITP and $(\aleph_3, \mu')$-ITP hold, for all $\mu\geq \aleph_2$ and $\mu'\geq \aleph_3.$
Solomon Feferman - One of the best experts on this subject based on the ideXlab platform.
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Operational set theory and small large Cardinals
Information and Computation, 2009Co-Authors: Solomon FefermanAbstract:A new axiomatic system OST of operational set theory is introduced in which the usual language of set theory is expanded to allow us to talk about (possibly partial) operations applicable both to sets and to operations. OST is equivalent in strength to admissible set theory, and a natural extension of OST is equivalent in strength to ZFC. The language of OST provides a framework in which to express ''small'' large Cardinal notions-such as those of being an Inaccessible Cardinal, a Mahlo Cardinal, and a weakly compact Cardinal-in terms of operational closure conditions that specialize to the analogue notions on admissible sets. This illustrates a wider program whose aim is to provide a common framework for analogues of large Cardinal notions that have appeared in admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics, and systems of recursive ordinal notations that have been used in proof theory.
Jaykov Foukzon - One of the best experts on this subject based on the ideXlab platform.
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Consistency results in topology and homotopy theory
2016Co-Authors: Jaykov FoukzonAbstract:let κ be an Inaccessible Cardinal and Hk is a set of all sets having hereditary size less then κ, then Con(ZFC + (V = Hk)), (2) there is a Lindelöf T3 indestructible space of pseudocharacter ≤N1 and size N2 in
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Inconsistent Countable Set in Second Order ZFC and Nonexistence of the Strongly Inaccessible Cardinals
British Journal of Mathematics & Computer Science, 2015Co-Authors: Jaykov FoukzonAbstract:In this article we derived an importent example of the inconsistentcountable set in second order ZFC (ZFC2) with the full second-order semantic.Main results is: (i) ¬Con(ZFC2), (ii) let k be an Inaccessible Cardinal and Hk is a set of all sets having hereditary size less then k, then ¬Con(ZFC + (V = Hk)).
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Consistency Results in Topology and Homotopy Theory
Pure and Applied Mathematics Journal, 2015Co-Authors: Jaykov FoukzonAbstract:Main results is: (1) let κ be an Inaccessible Cardinal and Hk is a set of all sets having hereditary size less then κ, then Con(ZFC + (V = Hk )), (2) there is a Lindelof T3 indestructible space of pseudocharacter ≤N1 and size N2 in L.
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Generalized Lob's Theorem.Strong Reflection Principles and Large Cardinal Axioms.Consistency Results in Topology
Advances in Pure Mathematics, 2013Co-Authors: Jaykov FoukzonAbstract:In this article, a possible generalization of the Lob’s theorem is considered. Main result is: let κ be an Inaccessible Cardinal, then
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There is No Standard Model of ZFC and ZFC_2 with Henkin semantics.Generalized Lob's Theorem.Strong Reflection Principles and Large Cardinal Axioms.Consistency Results in Topology.
arXiv: General Mathematics, 2013Co-Authors: Jaykov FoukzonAbstract:In this article we proved so-called strong reflection principles corresponding to formal theories Th which has omega-models. An posible generalization of the Lob's theorem is considered.Main results is: (1) let $k$ be an Inaccessible Cardinal, then $\neg Con(ZFC+\exists k)$,(2) there is a Lindel\"of $T_3$ indestructible space of pseudocharacter $\leqslant \aleph_1$ and size $\aleph_2$ in $L$.
Dima Sinapova - One of the best experts on this subject based on the ideXlab platform.
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The super tree property at the successor of a singular
arXiv: Logic, 2018Co-Authors: Sherwood Hachtman, Dima SinapovaAbstract:For an Inaccessible Cardinal $\kappa$, the super tree property (ITP) at $\kappa$ holds if and only if $\kappa$ is supercomact. However, just like the tree property, it can hold at successor Cardinals. We show that ITP holds at the successor of the limit of $\omega$ many supercompact Cardinals. Then we show that it can consistently hold at $\aleph_{\omega+1}$. We also consider a stronger principle, ISP, and certain weaker variations of it. We determine which level of ISP can hold at a successor of a singular. These results fit in the broad program of testing how much compactness can exist in the universe, and obtaining large Cardinal-type properties at smaller Cardinals.
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SINGULAR CardinalS AND SQUARE PROPERTIES
2014Co-Authors: Menachem Magidor, Dima SinapovaAbstract:Abstract. We analyze the effect of singularizing Cardinals on square properties. An old theorem of Dzamonja-Shelah/Gitik says that if you singularize an Inaccessible Cardinal while preserving its successor, then κ,ω holds in the bigger model. We extend this to the situation where a finite interval of Cardinals above κ is collapsed. More precisely, we show that if V ⊂ W, κ is Inaccessible in V, cfW (κ+iV) = ω for all 0 ≤ i ≤ n, and κ+n+1V = κ W, then W | = κ,ω
