Covering of the null ideal may have countable cofinality
We prove that it is consistent that the covering of the ideal of measure zero sets has countable cofinality.
math.LO↗
arXiv subjects
Saharon Shelah
Publications and source records attributed to Saharon Shelah.
There are no infinite order polynomially complete lattices after all
A lattice L is called opc if every monotone function f : L^n -> L is induced by a polynomial. We show here: If L is a lattice with the interpolation property whose cardinality is a strong limit cardinal of uncountable cofinality, then some finite power L^n has an antichain of size kappa. Using our previous result (math.LO/9707203 in the xxx archive) that the cardinality of an infinite opc lattice must be inaccessible, we can now conclude that there are no infinite opc lattices. However, the existence of strongly amorphous sets implies (in ZF) the existence of infinite opc lattices.
math.LO↗
On Hanf numbers of the infinitary order property
We study several cardinal, and ordinal--valued functions that are relatives of Hanf numbers. Let kappa be an infinite cardinal, and let T subseteq L_{kappa^+, omega} be a theory of cardinality <= kappa, and let gamma be an ordinal >= kappa^+. For example we look at (1) mu_{T}^*(gamma, kappa):= min {mu^* for all phi in L_{infinity, omega}, with rk(phi)< gamma, if T has the (phi, mu^*)-order property then there exists a formula phi'(x;y) in L_{kappa^+, omega}, such that for every chi >= kappa, T has the (phi', chi)-order property}; and (2) mu^*(gamma, kappa):= sup{mu_T^*(gamma, kappa)| T in L_{kappa^+,omega}}.
math.LO↗
Categoricity of abstract classes with amalgamation
Let K be an abstract elementary class with amalgamation, and Lowenheim Skolem number LS(K). We prove that for a suitable Hanf number chi_0 if chi_0 < lambda_0 <= lambda_1, and K is categorical in lambda^+_1 then it is categorical in lambda_0 .
math.LO↗
Borel Whitehead groups
We investigate the Whiteheadness of Borel abelian groups (aleph_1-free, wlog, as otherwise this is trivial). We show that CH (and even WCH) implies any such abelian group is free, and always aleph_2-free.
math.LO↗
The Generalized Continuum Hypothesis revisited
We argue that we solved Hilbert's first problem positively (after reformulating it just to avoid the known consistency results) and give some applications. Let lambda to the revised power of kappa, denoted lambda^{[kappa]}, be the minimal cardinality of a family of subsets of lambda each of cardinality kappa such that any other subset of lambda of cardinality kappa is included in the union of aleph_0, (for all theta = mu for some kappa lambda^{[theta]}= lambda and (b) there is a family P of lambda subsets of lambda each of cardinality < mu such that every subset of lambda of cardinality mu is equal to the union of < kappa members of P .
math.LO↗
On quantification with a finite universe
We consider a finite universe U (more exactly - a family U of them) and second order quantifiers Q_K, where for each U this means quantifying over a family of n(K)-place relations closed under permuting U. We define some natural orders and shed some light on the classification problem of those quantifiers.
math.LO↗
Universal graphs with forbidden subgraphs and algebraic closure
We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an associated''algebraic closure'' operator. The main applications are new examples of universal graphs with forbidden subgraphs and simplified treatments of some previously known cases.
math.LO↗
On reflection of stationary sets in P_kappa lambda
We investigate reflection of stationary sets in P_kappa lambda and prove a consistency result for the case when lambda is the successor of kappa.
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Non-existence of universal members in classes of abelian groups
We prove that if mu^+< lambda =cf(lambda)< mu^{aleph_0}, then there is no universal reduced torsion free abelian group. Similarly if aleph_0< lambda < 2^{aleph_0}. We also prove that if 2^{aleph_0}< mu^+< lambda =cf(lambda)< mu^{aleph_0}, then there is no universal reduced separable abelian p-group in lambda. (Note: both results fail if lambda = lambda^{aleph_0} or if lambda is strong limit, cf (mu)= aleph_0< mu).
math.LO↗
Not collapsing cardinals <= kappa in (< kappa) --support iterations II
We deal with (< kappa)-supported iterated forcing notions which are (E_0,E_1)-complete, have in mind problems on Whitehead groups, uniformizations and the general problem. We deal mainly with the successor of a singular case. This continues [Sh:587]. We also deal with complimentary combinatorial results.
math.LO↗
More on cardinal invariants of Boolean algebras
We address several questions of Donald Monk related to irredundance and spread of Boolean algebras, gaining both some ZFC knowledge and consistency results. We show in ZFC that irr(B_0 times B_1)= max(irr(B_0),irr(B_1)). We prove consistency of the statement ``there is a Boolean algebra B such that irr(B) Aut(B_1). Finally we show that consistently there is a Boolean algebra B of size lambda such that there is no free sequence in B of length lambda, there is an ultrafilter of tightness lambda (so t(B)=lambda) and lambda notin Depth_(Hs)(B).
math.LO↗
Norms on possibilities I: forcing with trees and creatures
We present a systematic study of the method of "norms on possibilities" of building forcing notions with keeping their properties under full control. This technique allows us to answer several open problems, but on our way to get the solutions we develop various ideas interesting per se.These include a new iterable condition for ``not adding Cohen reals'' (which has a flavour of preserving special properties of p-points), new intriguing properties of ultrafilters (weaker than being Ramsey but stronger than p-point) and some new applications of variants of the PP--property.
math.LO↗
PCF and infinite free subsets
We give another proof that for every lambda >= beth_omega for every large enough regular kappa < beth_omega we have lambda^{[kappa]}= lambda, dealing with sufficient conditions for replacing beth_omega by aleph_omega. In section 2 we show that large pcf(a) implies existence of free sets. An example is that if pp(aleph_omega)> aleph_{omega_1} then for every algebra M of cardinality aleph_omega with countably many functions, for some a_n in M (for n< omega) we have a_n notin cl_M({a_l: l not= n, l ) and more general version, and from assumptions as in section 2 get results even for the non stationary ideal. Lastly, we deal with some other measurements of [lambda]^{>= theta} and give an application by a construction of a Boolean Algebra.
math.LO↗
Lusin sequences under CH and under Martin's Axiom
Assuming the continuum hypothesis there is an inseparable sequence of length omega_1 that contains no Lusin subsequence, while if Martin's Axiom and the negation of CH is assumed then every inseparable sequence (of length omega_1) is a union of countably many Lusin subsequences.
math.LO↗
Polynomial time logic: Inability to express
Here we deal with the logic of [GuSh 533], which tries to capture polynomial time (for finite models). There it is proved that the logic cannot say much on models with equality only. Here we prove that it cannot say much on models for which we expect it cannot say much, like random enough graphs. This is the result of having a general criterion.
math.LO↗
On inverse gamma-systems and the number of L_{infty,lambda}-equivalent, non-isomorphic models for lambda singular
Suppose lambda is a singular cardinal of uncountable cofinality kappa. For a model M of cardinality lambda, let No(M) denote the number of isomorphism types of models N of cardinality lambda which are L_{infty lambda}-equivalent to M. In [Sh:189] inverse kappa-systems A of abelian groups and their certain kind of quotient limits Gr(A)/Fact(A) were considered. It was proved that for every cardinal mu there exists an inverse kappa-system A such that A consists of abelian groups having cardinality at most mu^kappa and card(Gr(A)/Fact(A))= mu. In [Sh:228] a strict connection between inverse kappa-systems and possible values of No was proved. In this paper we show: for every nonzero mu <= lambda^kappa there is an inverse kappa-system A of abelian groups having cardinality < lambda such that card(Gr(A)/Fact(A))= mu (under the assumptions 2^kappa < lambda and theta^{< kappa}< lambda for all theta < lambda when mu > lambda), with the obvious new consequence concerning the possible value of No. Specifically, the case No(M)= lambda is possible when theta^kappa < lambda for every theta < lambda.
math.LO↗
Consistency of the existence of a 4-place function on aleph_2 with no independent subset of size k_n
It is consistent that there is a set mapping from the four-tuples of omega_n into the finite subsets with no free subsets of size t_n for some natural number t_n. For any n< omega it is consistent that there is a set mapping from the pairs of omega_n into the finite subsets with no infinite free sets.
math.LO↗