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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 163 records · Page 9Linked to original sources
arXiv2020

Controlling classical cardinal characteristics while collapsing cardinals

Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we show how we can compose $P$ with a collapse (of a cardinal $λ>κ$ to $κ$) such that the composition still forces the previous values to these characteristics. We also show how to force distinct values to $\mathfrak m$, $\mathfrak p$ and $\mathfrak h$ and also keeping all the values in Cichoń's diagram distint, using the Boolean Ultrapower method of arXiv:1708.03691 . (In arXiv:2006.09826 , the same was done for the newer Cichoń's Maximum construction, which avoids large cardinals.)

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arXiv2020

Cichoń's maximum without large cardinals

Cichoń's diagram lists twelve cardinal characteristics (and the provable inequalities between them) associated with the ideals of null sets, meager sets, countable sets, and $σ$-compact subsets of the irrationals. It is consistent that all entries of Cichoń's diagram are pairwise different (apart from $\textrm{add}(\mathcal{M})$ and $\textrm{cof}(\mathcal{M})$, which are provably equal to other entries). However, the consistency proofs so far required large cardinal assumptions. In this work, we show the consistency without such assumptions.

math.LO↗
arXiv2020

Turing Invariant Sets and the Perfect Set Property

We show that $ZF+DC+$"all Turing invariant sets of reals have the perfect set property" implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations.

math.LO↗
arXiv2020

Specializing trees and answer to a question of Williams

We show that if $cf(2^{\aleph_0})=\aleph_1,$ then any non-trivial $\aleph_1$-closed forcing notion of size $\leq 2^{\aleph_0}$ is forcing equivalent to $Add(\aleph_1, 1),$ the Cohen forcing for adding a new Cohen subset of $ω_1.$ We also produce, relative to the existence of suitable large cardinals, a model of $ZFC$ in which $2^{\aleph_0}=\aleph_2$ and all $\aleph_1$-closed forcing notion of size $\leq 2^{\aleph_0}$ collapse $\aleph_2,$ and hence are forcing equivalent to $Add(\aleph_1, 1).$ These results answer a question of Scott Williams from 1978. We also extend a result of Todorcevic and Foreman-Magidor-Shelah by showing that it is consistent that every partial order which adds a new subset of $\aleph_2,$ collapses $\aleph_2$ or $\aleph_3.$

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arXiv2020

On $κ$-homogeneous, but not $κ$-transitive permutation groups

A permutation group $G$ on a set $A$ is $κ$-homogeneous iff for all $X,Y\in [A]^κ$ with $|A\setminus X|=|A\setminus Y|=|A|$ there is a $g\in G$ with $g[X]=Y$. $G$ is $κ$-transitive iff for any injective function $f$ with $dom(f)\cup ran(f)\in [A]^{\le κ}$ and $|A\setminus dom(f)|=|A\setminus ran(f)|=|A|$ there is a $g\in G$ with $f\subset g$. Giving a partial answer to a question of P. M. Neumann we show that there is an $ω$-homogeneous but not $ω$-transitive permutation group on a cardinal $λ$ provided (i) $λ<ω_ω$, or (ii) $2^ω<λ$, and $μ^ω=μ^+$ and $\Box_μ$ hold for each $μ\leλ$ with $ω=cf(μ)<{μ}$, or (iii) our model was obtained by adding $ω_1$ many Cohen generic reals to some ground model. For $κ>ω$ we give a method to construct large $κ$-homogeneous, but not $κ$-transitive permutation groups. Using this method we show that there exists $κ^+$-homogeneous, but not $κ^+$-transitive permutation groups on $κ^{+n}$ for each infinite cardinal $κ$ and natural number $n\ge 1$ provided $V=L$.

math.LO↗
arXiv2020

On CON(${\mathfrak d}_λ>$ cov$_λ$(meagre))

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.

math.LO↗
arXiv2020

Double weakness

We prove that, consistently, there exists a weakly but not strongly inaccessible cardinal $λ$ for which the sequence $\langle 2^θ:θ<λ\rangle$ is not eventually constant and the weak diamond fails at $λ$. We also prove that consistently diamond fails but a parametrized version of weak diamond holds at some strongly inaccessible $λ$.

math.LO↗
arXiv2019

Depth^+ and Length^+ of Boolean algebras

Let inv denote the cardinal invariants Depth^+ and Length^+ on Boolean algebras. For many singular cardinals we create a strict inequality between the product of the inv values and the inv of the product algebra. The proof holds in ZFC.

math.RA↗
arXiv2019

Some Results on Polish Groups

We prove that no quantifier-free formula in the language of group theory can define the $\aleph_1$-half graph in a Polish group, thus generalising some results from [6]. We then pose some questions on the space of groups of automorphisms of a given Borel complete class, and observe that this space must contain at least one uncountable group. Finally, we prove some results on the structure of the group of automorphisms of a locally finite group: firstly, we prove that it is not the case that every group of automorphisms of a graph of power $λ$ is the group of automorphism of a locally finite group of power $λ$; secondly, we conjecture that the group of automorphisms of a locally finite group of power $λ$ has a locally finite subgroup of power $λ$, and reduce the problem to a problem on $p$-groups, thus settling the conjecture in the case $λ= \aleph_0$.

math.LO↗
arXiv2019

Indecomposable explicit abelian group

(withdrawn.) For every lambda we give an explicit construction of an Abelian group with no non-trivial automorphisms. In particular the group absolutely has no non-trivial automorphisms, hence is absolutely indecomposable. Earlier we knew a stronger existence theorem but only up to a quite large cardinal which was a necessary restriction. In another direction the construction does not use the axiom of choice.

math.LO↗
arXiv2019

$\aleph_k$-free cogenerators

We prove in ZFC that an abelian group $C$ is cotorsion if and only if $\operatorname{Ext}(F,C) = 0$ for every $\aleph_k$-free group $F$, and discuss some consequences and related results. This short note includes a condensed overview of the $\barλ$-Black Box for $\aleph_k$-free constructions in ZFC.

math.GR↗
arXiv2019

Distorted sums of models

Distorted sums of models were introduced and discussed in [Sh:463]. This notion generalizes the notion of disjoint (or direct) sums of models by letting the summands overlap. In the first section we investigate types in distorted sums and show that the type of a sequence of elements A is determined by the `local' type of A (i.e. the type restricted to a neighborhood of A). We simplify the proofs in [Sh:463] and improve the bounds on the radii needed to determine the types. Natural examples of distorted sums are models with distant functions. In the second and third sections we discuss such models and improve a theorem by Gaifman, that states that each formula is equivalent to a boolean combination of local formulas.

math.LO↗
arXiv2019

Adding a random real number and its effect on Martin's axiom

We show that adding a random real number destroys a large fragment of Martin's axiom, namely Martin's axiom for partial orders that have precalibre-$\aleph_1$, thus answering an old question of J. Roitman [9]. We also answer a question of J. Steprans and S. Watson [13] by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-$\aleph_1$ property of a partial ordering while preserving its ccc-ness.

math.LO↗
arXiv2019

Club Chang's Conjecture

Chang's Conjecture (CC) asserts that for every $F:[ω_2]^{<ω} \to ω_2$, there exists an $X$ that is closed under $F$ such that $|X|=ω_1$ and $|X \cap ω_1| =ω$. By classic results of Silver and Donder, CC is equiconsistent with an $ω_1$-Erdos cardinal. Using stronger large cardinal assumptions (between $o(κ) = κ^+$ and $o(κ) = κ^{++}$), we prove that it is consistent to also require that $X$ contains a closed unbounded set of ordinals in $\text{sup}(X \cap ω_2)$. We denote this stronger principle \textbf{Club-CC}, and also show that, unlike CC, Club-CC implies failure of certain weak square principles.

math.LO↗
arXiv2019

There is a maximal homogeneous family over $ω$

We tried to show in this paper that CH implies: there is a homogeneous family of subset of omega which is maximal with respect to inclusion. The paper is currently "withdrawn", since a mistake has been discovered.

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