Searcharxiv⌕ Search

arXiv subjects

Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 361 records · Page 20Linked to original sources
arXiv2010

Creature forcing and large continuum: The joy of halving

For $f,g\inω^ω$ let $c^\forall_{f,g}$ be the minimal number of uniform $g$-splitting trees needed to cover the uniform $f$-splitting tree, i.e., for every branch $ν$ of the $f$-tree, one of the $g$-trees contains $ν$. Let $c^\exists_{f,g}$ be the dual notion: For every branch $ν$, one of the $g$-trees guesses $ν(m)$ infinitely often. We show that it is consistent that $c^\exists_{f_ε,g_ε}=c^\forall_{f_ε,g_ε}=κ_ε$ for continuum many pairwise different cardinals $κ_ε$ and suitable pairs $(f_ε,g_ε)$. For the proof we introduce a new mixed-limit creature forcing construction.

math.LO↗
arXiv2010

What majority decisions are possible with possible abstaining

Suppose we are given a family of choice functions on pairs from a given finite set. The set is considered as a set of alternatives (say candidates for an office) and the functions as potential "voters". The question is, what choice functions agree, on every pair, with the majority of some finite subfamily of the voters? For the problem as stated, a complete characterization was given in \citet{shelah2009mdp}, but here we allow each voter to abstain. There are four cases.

math.CO↗
arXiv2010

Splitting stationary sets from weak forms of Choice

Working in the context of restricted forms of the Axiom of Choice, we consider the problem of splitting the ordinals below $λ$ of cofinality $θ$ into $λ$ many stationary sets, where $θ< λ$ are regular cardinals. This is a continuation of \cite{Sh835}.

math.LO↗
arXiv2010

Universally measurable sets in generic extensions

A subset of a topological space is said to be \emph{universally measurable} if it is measured by the completion of each countably additive $σ$-finite Borel measure on the space, and \emph{universally null} if it has measure zero for each such atomless measure. In 1908, Hausdorff proved that there exist universally null sets of real numbers of cardinality $\aleph_{1}$, and thus that there exist at least $2^{\aleph_{1}}$ such sets. Laver showed in the 1970's that consistently there are just continuum many universally null sets of reals. The question of whether there exist more than continuum many universally measurable sets of reals was asked by Mauldin in 1978. We show that consistently there exist only continuum many universally measurable sets. This result also follows from work of Ciesielski and Pawlikowski on the iterated Sacks model. In the models we consider (forcing extensions by suitably-sized random algebras) every set of reals is universally measurable if and only if it and its complement are unions of ground model continuum many Borel sets.

math.LO↗
arXiv2010

The Stationary Set Splitting Game

The \emph{stationary set splitting game} is a game of perfect information of length $ω_{1}$ between two players, \unspls and \spl, in which \unspls chooses stationarily many countable ordinals and \spls tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to $Σ^{2}_{2}$ maximality with a predicate for the nonstationary ideal on $ω_{1}$, and an example of a consistently undetermined game of length $ω_{1}$ with payoff definable in the second-order monadic logic of order. We also show that the determinacy of the game is consistent with Martin's Axiom but not Martin's Maximum.

math.LO↗
arXiv2010

Large continuum, oracles

Our main theorem is about iterated forcing for making the continuum larger than aleph_2. We present a generalization of math.LO/0303294 which is dealing with oracles for random, etc., replacing aleph_1, aleph_2 by lambda,lambda^+ (starting with lambda=lambda^{ aleph_1). Well, instead of properness we demand absolute c.c.c. So we get, e.g. the continuum is lambda^+ but we can get cov(meagre)=lambda. We give some applications. As in math.LO/0303294, it is a "partial" countable support iteration but it is c.c.c.

math.LO↗
arXiv2010

PCF arithmetic without and with choice

We deal with relatives of GCH which are provable. In particular we deal with rank version of the revised GCH. Our motivation was to find such results when only weak versions of the axiom of choice are assumed but some of the results gives us additional information even in ZFC.

math.LO↗
arXiv2010

Good Frames With A Weak Stability

Let K be an abstract elementary class of models. Assume that there are less than the maximal number of models in K_{λ^{+n}} (namely models in K of power λ^{+n}) for all n. We provide conditions on K_λ, that imply the existence of a model in K_{λ^{+n}} for all n. We do this by providing sufficiently strong conditions on K_λ, that they are inherited by a properly chosen subclass of K_{λ^+}.

math.LO↗
arXiv2009

A dichotomy for the number of ultrapowers

We prove a strong dichotomy for the number of ultrapowers of a given countable model associated with nonprincipal ultrafilters on N. They are either all isomorphic, or else there are $2^{2^{\aleph_0}}$ many nonisomorphic ultrapowers. We prove the analogous result for metric structures, including C*-algebras and II$_1$ factors, as well as their relative commutants and include several applications. We also show that the C*-algebra B(H) always has nonisomorphic relative commutants in its ultrapowers associated with nonprincipal ultrafilters on N.

math.LO↗
arXiv2009

Stable theories and representation over sets

In this paper we explore the representation property over sets. This property generalizes constructibility, however is weak enough to enable us to prove that the class of theories $T$ whose models are representable is exactly the class of stable theories. Stronger results are given for omega-stable.

math.LO↗
arXiv2009

Uniform n-place functions on T\subseteq ds(α)

In this paper the Erdos-Rado theorem is generalized to the class of well founded trees. We define an equivalence relation on the class rs(infty)^{< aleph_0} (finite sequences of decreasing sequences of ordinals) with aleph_0 equivalence classes, and for n< omega a notion of n-end-uniformity for a colouring of rs(infty)^{< aleph_0} with mu colours. We then show that for every ordinal alpha, n< omega and cardinal mu there is an ordinal lambda so that for any colouring c of T=rs(lambda)^{< aleph_0} with mu colours, T contains S isomorphic to rs(alpha) so that c rest S^{< aleph_0} is n-end uniform. For c with domain T^n this is equivalent to finding S subseteq T isomorphic to rs(alpha) so that c upharpoonright S^{n} depends only on the equivalence class of the defined relation, so in particular T-> (rs(alpha))^n_{mu, aleph_0} . We also draw a conclusion on colourings of n-tuples from a scattered linear order.

math.LO↗
arXiv2009

A Sacks Real out of Nowhere

There is a proper countable support iteration of length $ω$ adding no new reals at finite stages and adding a Sacks real in the limit.

math.LO↗
arXiv2009

Introduction to: classification theory for abstract elementary class

Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to find dividing lines, prove their consequences, prove "structure theorems, positive theorems" on those in the "low side" (in particular stable and superstable theories), and prove "non-structure, complexity theorems" on the "high side". It has started with categoricity and number of non-isomorphic models. It is probably recognized as the central part of model theory, however it will be even better to have such (non-trivial) theory for non-elementary classes. Note also that many classes of structures considered in algebra are not first order; some families of such classes are close to first order (say have kind of compactness). But here we shall deal with a classification theory for the more general case without assuming knowledge of the first order case (and in most parts not assuming knowledge of model theory at all). The present paper includes an introduction to the forthcoming book on Classification Theory for Abstract Elementary Classes

math.LO↗
arXiv2009

Strongly dependent theories

We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing math.LO/0406440. If |A|+|T|<= mu, I subseteq C, |I| >=beth_{|T|^+}(mu) then some J subseteq I of cardinality mu^+ is an indiscernible sequence over A .

math.LO↗
arXiv2008

Non-structure in lambda^{++} using instances of WGCH

We try to redo, improve and continue the non-structure parts in some works on a.e.c., which uses weak diamond, in lambda^+ and lambda^{++} getting better and more results and do what is necessary for the book on a.e.c. Comparing with math.LO/9805146 we make the context closer to the examples, hence hopefully improve transparency, though losing some generality. Toward this we work also on the positive theory, i.e. structure side of "low frameworks" like almost good lambda-frames.

math.LO↗
arXiv2008

Categoricity and solvability of A.E.C., quite highly

We investigate in ZFC what can be the family of large enough cardinals mu in which an a.e.c. K is categorical or even just solvable. We show that for not few cardinals lambda<mu there is a superlimit model in K_lambda. Moreover, our main result is that we can find a good lambda-frame s categorical in lambda such that K_s subseteq K_lambda. We then show how to use [Sh:705] to get categoricity in every large enough cardinality if K has cases of mu-amalgamation for enough mu and 2^mu<2^{mu^{+1}} <... < 2^{mu^{+n}}... for enough mu.

math.LO↗