What majority decisions are possible

The main result is the following: Let X be a finite set and D be a non empty family of choice functions for (X choose 2) closed under permutation of X. Then the following conditions are equivalent: (A) for any choice function c on (X choose 2) we can find a finite set J and c_j in D for j in J such that for any x not= y in X : c{x,y}=y Leftrightarrow |J|/2<| {j in J:c_j{x,y}= y}| (so equality never occurs) (B) for some c in D and x in X we have |{y: c{x,y}=y}| not= (|X|-1)/2 . We then describe what is the closure of a set of choice functions by majority; in fact, there are just two possibilities (in section 3). In section 4 we discuss a generalization

Quite Complete Real Closed fields

We prove that any ordered field can be extended to one for which every decreasing sequence of bounded closed intervals, of any length, has a nonempty intersection; equivalently, there are no Dedekind cuts with equal cofinality from both sides. Here we strengthen the results from the published version

Superatomic Boolean Algebras: maximal rigidity

We prove that for any superatomic Boolean Algebra of cardinality >beth_omega there is an automorphism moving uncountably many atoms. Similarly for larger cardinals. Any of those results are essentially best possible

Categoricity of an abstract elementary class in two successive cardinals

We investigate categoricity of abstract elementary classes without any remnants of compactness (like non-definability of well ordering, existence of E.M. models or existence of large cardinals). We prove (assuming a weak version of GCH around lambda) that if K is categorical in lambda, lambda^+, LS(K) <= lambda and 1 <= I(lambda^{++},K)< 2^{lambda^{++}} then K has a model in lambda^{+++}

Characterizing aleph_epsilon-saturated models of superstable ndop theories by L_{infty, aleph_epsilon}-theory

After the main gap theorem was proved (see [Sh:c]), in discussion, Harrington expressed a desire for a finer structure - of finitary character (when we have a structure theorem at all). I point out that the logic L_{infty,aleph_0}(d.q.) (d.q. stands for dimension quantifier) does not suffice: e.g., for T=Th(lambda x 2^\omega,E_n)_{n<omega} where (alpha,eta)E_n(beta,nu) =: eta|n=nu|n and for a subset S of 2^omega we define M_S = M | {(alpha,eta): [eta in S -> alpha<omega_1] and [eta in 2^\omega backslash S -> alpha<omega]}. Hence, it seems to me we should try L_{infty,aleph_epsilon}(d.q.) (essentially, in C we can quantify over sets which are included in the algebraic closure of finite sets), and Harrington accepts this interpretation. Here the conjecture is proved for aleph_epsilon-saturated models. I.e., the main theorem is M equiv_{L_{infty,aleph_epsilon}(d.q.)} N iff M cong N for aleph_epsilon--saturated models of a superstable countable (first order) theory T without dop

Universal in (< lambda)-stable abelian group

A characteristic result is that if 2^{aleph_0}< mu < mu^+< lambda = cf(lambda)< mu^{aleph_0}, then among the separable reduced p-groups of cardinality lambda which are (< lambda)-stable there is no universal one

On full Souslin trees

In this note we answer a question of Kunen (15.13 [Mi91]) showing that it is consistent that there are full Souslin tree

There may be no nowhere dense ultrafilter

We show the consistency of ZFC +''there is no NWD-ultrafilter on omega'', which means: for every non principle ultrafilter D on the set of natural numbers, there is a function f from the set of natural numbers to the reals, such that for some nowhere dense set A of reals, the set {n: f(n) in A} is not in D. This answers a question of van Douwen, which was put in more general context by Baumgartne

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

Can groupwise density be much bigger than the non-dominating number?

We prove that g (the groupwise density number) is smaller or equal to b^+ (the successor of the minimal cardinality of a non-dominated subset of omega^omega)