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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 325 records · Page 18Linked to original sources
arXiv2011

A closed algebra with a non-Borel clone and an ideal with a Borel clone

Algebras on the natural numbers and their clones of term operations can be classified according to their descriptive complexity. We give an example of a closed algebra which has only unary operations and whose clone of term operations is not Borel. Moreover, we provide an example of a coatom in the clone lattice whose obvious definition via an ideal of subsets of natural numbers would suggest that it is complete coanalytic, but which turns out to be a rather simple Borel set.

math.RA↗
arXiv2011

Automorphism towers and automorphism groups of fields without Choice

This paper can be viewed as a continuation of [KS09] that dealt with the automorphism tower problem without Choice. Here we deal with the inequation which connects the automorphism tower and the normalizer tower without Choice and introduce a new proof to a theorem of Fried and Kollár that any group can be represented as an automorphism group of a field. The proof uses a simple construction: working more in graph theory, and less in algebra.

math.LO↗
arXiv2011

Universality of the lattice of transformation monoids

The set of all transformation monoids on a fixed set of infinite cardinality λ, equipped with the order of inclusion, forms a complete algebraic lattice Mon(λ) with 2^λ compact elements. We show that this lattice is universal with respect to closed sublattices, i.e., the closed sublattices of Mon(λ) are, up to isomorphism, precisely the complete algebraic lattices with at most 2^λ compact elements.

math.RA↗
arXiv2011

Comparing the closed almost disjointness and dominating numbers

We prove that if there is a dominating family of size ${\aleph}_{1}$, then there is are ${\aleph}_{1}$ many compact subsets of $ω^ω$ whose union is a maximal almost disjoint family of functions that is also maximal with respect to infinite partial functions.

math.LO↗
arXiv2011

Nonproper Products

We show that there exist two proper creature forcings having a simple (Borel) definition, whose product is not proper. We also give a new condition ensuring properness of some forcings with norms.

math.LO↗
arXiv2011

Clones above the unary clone

Let c be the cardinality of the continuum. We give a family of pairwise incomparable clones (on a countable base set) 2^c members, all with the same unary fragment, namely the set of all unary operations. We also give, for each n, a family of 2^c clones all with the same n-ary fragment, and all containing the set of all unary operations.

math.RA↗
arXiv2011

The character spectrum of beta(N)

We show the consistency of: the set of regular cardinals which are the character of some ultrafilter on omega can be quite chaotic, in particular not only can be not convex but can have many gaps. We also deal with the set of pi-characters of ultrafilters on omega.

math.GN↗
arXiv2011

The first almost free Whitehead group

Assume G.C.H. and kappa is the first uncountable cardinal such that there is a kappa-free abelian group which is not a Whitehead (abelian) group. We prove that kappa is necessarily an inaccessible cardinal

math.LO↗
arXiv2011

Nice infinitary logics

Ordinary infinitary languages L_{lambda, kappa} satisfy the Interpolation Theorem only in the case lambda <= {aleph_1}, kappa = {aleph_0}, this include first order logic of course. There are also some pairs of such logics satifying interpolation, e.g. (L_{lambda^+,{aleph_0}}, L_{(2^lambda)^+, lambda^+}) . Does this come from an intermidiate logic satisfying it? Is it nice? unique? We define for kappa = beth_kappa a new logic L^1_kappa such that L_{kappa omega}< L^1_kappa LL_{kappa kappa} and L^1_kappa is very nice; in particular satisfies the Interpolation Theorem. Moreover, L^1_kappa has a model--theoretic characterization in the style of Lindstrom's Theorem in terms of a form of undefinability of well--order. We also define for strong limit kappa of cofinality aleph_0 a logic L^2_{kappa^+} such that L_{kappa^+, {aleph_0}}<L^2_{kappa^+}<L_{kappa^+, kappa} and L^2_{kappa^+} satisfies the Interpolation Theorem.

math.LO↗
arXiv2011

Existence of Endo-Rigid Boolean Algebras

How many endomorphisms does a Boolean algebra have? Can we find Boolean algebras with as few endomorphisms as possible? Of course from any ultrafilter of the Boolean algebra we can define an endomorphism, and we can combine finitely many such endomorphisms in some reasonable ways. We prove that in any cardinality lambda=lambda^ {aleph_0} there is a Boolean algebra with no other endomorphisms. For this we use the so called "black boxes", but in a self contained way. We comment on how necessary the restriction on the cardinal is.

math.LO↗
arXiv2011

Weakening the local character

In [Sh E46], Shelah obtained a non-forking relation for an AEC, (K,\preceq), with LST-number at most λ, which is categorical in λand λ^+ and has less than 2^{λ^+} models of cardinality λ^{++}, but at least one. This non-forking relation satisfies the main properties of the non-forking relation on stable first order theories, but only a weak version of the local character. Here, we improve this non-forking relation such that it satisfies the local character, too. Therefore it satisfies the main properties of the non-forking relation on superstable first order theories. We conclude that the function λ\to I(λ,K), which assigns to each cardinal λ, the number of models in K of cardinality λ, is not arbitrary.

math.LO↗
arXiv2011

On triangleleft^*-maximality

This paper investigates a connection between the ordering triangleleft^ast among theories in model theory and the (N)SOP_n hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP_2 and SOP_1, and gives a strong connection between SOP_1 and the maximality in Keisler ordering. Together with the known results about the connection between the (N)SOP_n hierarchy and the existence of universal models in the absence of GCH, the paper provides a step toward the classification of unstable theories without the strict order property.

math.LO↗
arXiv2011

(kappa,theta)-weak normality

We characterize the situation of small cardinality for a product of cardinals divided by an ultrafilter. We develop the notion of weak normality. We include an application to Boolean Algebras.

math.LO↗
arXiv2011

Partition theorems from creatures and idempotent ultrafilters

We show a general scheme of Ramsey-type results for partitions of countable sets of finite functions, where "one piece is big" is interpreted in the language originating in creature forcing. The heart of our proofs follows Glazer's proof of the Hindman Theorem, so we prove the existence of idempotent ultrafilters with respect to suitable operation. Then we deduce partition theorems related to creature forcings.

math.LO↗
arXiv2011

Adding linear orders

We address the following question: Can we expand an NIP theory by adding a linear order such that the expansion is still NIP? Easily, if acl(A)=A for all A, then this is true. Otherwise, we give counterexamples. More precisely, there is a totally categorical theory for which every expansion by a linear order has IP. There is also an ω-stable NDOP theory for which every expansion by a linear order interprets bounded arithmetic.

math.LO↗
arXiv2011

Saccharinity

We present a method to iterate finitely splitting lim-sup tree forcings along non-wellfounded linear orders. We apply this method to construct a forcing (without using an inaccessible or amalgamation) that makes all definable sets of reals measurable with respect to a certain (non-ccc) ideal.

math.LO↗
arXiv2011

No limit model in inaccessible

Our aim is to improve the negative results i.e. non-existence of limit models, and the failure of the generic pair property from math.LO/0609636 to inaccessible lambda as promised there. The motivation is that in [Sh:F756] the positive results are for lambda measurable hence inaccessible, whereas in math.LO/0609636 in the negative results obtained only on non-strong limit cardinals.

math.LO↗