Every two elementarily equivalent models have isomorphic ultrapowers

We prove (without G.C.H.) that every two elementarily equivalent models have isomorphic ultrapowers, and some related results. The preparation of this paper was supported in Part by NSF Grant GP-22937.     

On two problems of Erdos and Hechler: New methods in singular madness

For an infinite cardinal , denotes the set of all cardinalities of nontrivial maximal almost disjoint families over .

Erdos and Hechler proved in 1973 the consistency of for a singular cardinal and asked if it was ever possible for a singular that , and also whether for every singular cardinal .

We introduce a new method for controlling for a singular and, among other new results about the structure of for singular , settle both problems affirmatively.

     

Antichains in Products of Linear Orders

We show that:(1) For many regular cardinals (in particular, for all successors of singular strong limit cardinals, and for all successors of singular -limits), for all n{2,3,4,...}: There is a linear order L such that L ⁿ has no (incomparability-)antichain of cardinality , while L ⁿ⁺¹ has an antichain of cardinality .(2) For any nondecreasing sequence _n :n{2,3,4,...} of infinite cardinals it is consistent that there is a linear order L such that, for all n: L ⁿ has an antichain of cardinality _n , but no antichain of cardinality _n ⁺.     

Perfectly meager sets and universally null sets

We will show that there is no example of a set distinguishing between universally null and perfectly meager sets.

     

Strongly almost disjoint sets and weakly uniform bases

A combinatorial principle CECA is formulated and its equivalence with GCH + certain weakenings of for singular is proved. CECA is used to show that certain ``almost point-' families can be refined to point- families by removing a small set from each member of the family. This theorem in turn is used to show the consistency of ``every first countable -space with a weakly uniform base has a point-countable base.'

     

Models with few isomorphic expansions

We shall characterize the countable modelsM, with only countably many expansions by a one place predicate. The second author is thankful for NSF Grant 43901, by which he was partially supported.     

Successor of singulars: Combinatorics and not collapsing cardinals ≤<Emphasis Type="Italic">gk</Emphasis> in (<<Emphasis Type="Italic">gk</Emphasis>)-support iterations

On the one hand, we deal with (<gk)-supported iterated forcing notions which are , bearing in mind problems on Whitehead groups, uniformizations and the general problem. We deal mainly with the case of a successor of the singular cardinal. This continues [Sh 587]. On the other hand, we deal with complimentary ZFC combinatorial results. I would like to thank Alice Leonhardt for the beautiful typing. This research was supported by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. Publication 667.     

A Countable Structure Does Not Have a Free Uncountable Automorphism Group

It is proved that the automorphism group of a countable structurecannot be a free uncountable group. The idea is that insteadof proving that every countable set of equations of a certainform has a solution, it is proved that this holds for a co-meagrefamily of appropriate countable sets of equations. This researchwas partially supported by the Israel Science Foundation.     

Infinite abelian groups,whitehead problem and some constructions

We solve here some problems from Fuchs’ book. We show that the answer to Whitehead’s problem (for groups of power ?₁) is independent from the usual axioms of set theory. We prove the existence of large rigid systems for groups of power λ, with no restriction on λ. We also prove that there are many non-isomorphic reduced separablep-groups.