Multi-dimensionality

We prove that ifT is stable, not multi-dimensional theory, then there is an infinite indiscernible set orthogonal to the empty set. This completes the proof that if ℵ_α=ℵ _•T• ^α >ℵ≥_Kr(T), thenT has ≥2^•α−β• non-isomorphic ℵ_β models of cardinality ℵ_α. Originally written November 5, 1988. Publication 429. Partially supported by the Israel-United States Binational Science Foundation; I thank Alice Leonhardt for the beautiful typing.     

ℵω may have a strong partition relation

We prove the consistency, with ZFC+G.C.H., of a strong partition relation of ℵω assuming the consistency of the existence of infinitely many compact cardinals. The author would like to thank the United States-Israel Binational Science Foundation for supporting this research by Grant No. 1110.     

Large k-preserving sets in infinite graphs

Let K be a cardinal. If K χ₀, define K := K . Otherwise, let K := K + 1. We prove a conjecture of Mader: Every infinite K -connected graph G = (V, E) contains a set S ? V with |S| = |V| such that G/S is K -connected for all S? S.     

“Gap 1” two-cardinal principles and the omitting types theorem for ℒ(Q)

Forλ a strong limit singular cardinal, and more generally forλ > 2^cofλ , we prove the equivalence of a number of model theoretic and combinatorial conditions, including the ℒ(Q)-completeness theorem for theλ ⁺-interpretation, an omitting types theorem for ℒ(Q) in theλ ⁺-interpretation, and a weak form of Jensen’s principle □_λ. Research supported in part by the United States-Israel Binational Science Foundation. The author thanks the referee for a thorough revision of the paper. Publication #269.     

On uncountable abelian groups

We continue the investigation from [10], [11], [12] on uncountable abelian groups. This paper tends more to group theory and was motivated by Nunke’s statement (in [9]) that Whitehead problem, rephrased properly, is not solved yet. The author would like to thank the NSF for partially supporting this research by grants 144-H747 and M2S76-08479, and the BSF for partially supporting this research by grant 1110.     

On measure and category

We show that under ZF + DC, even if every set of reals is measurable, not necessarily every set of reals has the Baire property. This was somewhat surprising, as for the Σ ₂ ¹ set the implication holds. This research was partially supported by an NSF grant and the U.S.-Israel Binational Science Foundation.     

Exponentiation in power series fields

We prove that for no nontrivial ordered abelian group does the ordered power series field admit an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a non-surjective logarithm. For an arbitrary ordered field , no exponential on is compatible, that is, induces an exponential on through the residue map. This is proved by showing that certain functional equations for lexicographic powers of ordered sets are not solvable.

     

On the consistency of some partition theorems for continuous colorings,and the structure of ℵ1-dense real order types

We present some techniques in c.c.c. forcing, and apply them to prove consistency results concerning the isomorphism and embeddability relations on the family of ?₁-dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two ?₁-dense subsets of $\text{R}$ are isomorphic, is consistent. We e.g. prove Con(BA+(2^?₀>?₂)). Let <K^H,<> be the set of order types of ?₁-dense homogeneous subsets of $\text{R}$ with the relation of embeddability. We prove that for every finite model <L, <->: Con(MA+ <K^H, <-> ? <L, <->) iff L is a distributive lattice. We prove that it is consistent that the Magidor-Malitz language is not countably compact. We deal with the consistency of certain topological partition theorems. E.g. We prove that MA is consistent with the axiom OCA which says: “If X is a second countable space of power ?₁, and {U₀,\h.;,U_n?1} is a cover of D(X)

XxX-}<x,x>¦x?X} consisting of symmetric open sets, then X can be partitioned into {X_i \brvbar; i ? ω} such that for every i ? ω there is l<n such that D(X_i)?U_l”. We also prove that MA+OCA [xrArr] 2 ?₀ = ?₂.     

Iterated forcing and changing cofinalities

We weaken the notion of proper to semi-proper, so that the important properties (e.g., being preserved by some interations) are preserved, and it includes some forcing which changes the confinality of a regular cardinal >ℵ₁ to ℵ₀. So, using the right iteractions, we can iterate such forcing without collapsing ℵ₁. As a result, we solve the following problems of Friedman, Magidor and Avraham, by proving (modulo large cardinals) the consistency of the following with G.C.H.: (1) for everyS ⊑ ℵ₂,S or ℵ₂-S contains a closed copy of ω₁ (2) there is a normal precipitous filterD on (3) for every is regular inL (δ ∩A)} is statonary. The results can be improved to equi-consistency; this will be discussed in a future paper. The author thanks the United States-Israel Binational Science Foundation for supporting the research by grant 1110.