A model of $$\mathsf {ZFA}+ \mathsf {PAC}$$ with no outer model of $$\mathsf {ZFAC}$$ZFAC with the same pure part

We produce a model of \(\mathsf {ZFA}+ \mathsf {PAC}\) such that no outer model of \(\mathsf {ZFAC}\) has the same pure sets, answering a question asked privately by Eric Hall.     

On the cofinality of the splitting number

The splitting number $\mathfrak{s}$ can be singular. The key method is to construct a forcing poset with finite support matrix iterations of ccc posets introduced by Blass and Shelah (1989).     

Polish topologies for graph products of cyclic groups

We give a complete characterization of the graph products of cyclic groups admitting a Polish group topology, and show that they are all realizable as the group of automorphisms of a countable structure. In particular, we characterize the right-angled Coxeter groups (resp. Artin groups) admitting a Polish group topology. This generalizes results from [8], [9] and [4].     

Further cardinal arithmetic

We continue the investigations in the author’s book on cardinal arithmetic, assuming some knowledge of it. We deal with the cofinality of (S _{≤ℵ 0}(κ), ⊆) for κ real valued measurable (Section 3), densities of box products (Section 5,3), prove the equality cov(λ,λ,θ⁺,2) in more cases even when cf(λ)=ℵ₀ (Section 1), deal with bounds of pp(λ) for λ limit of inaccessible (Section 4) and give proofs to various claims I was sure I had already written but did not find (Section 6). Done mainly 1–4/1991. I thank Alice Leonhardt for typing and retyping so beautifully and accurately. Partially supported by the Basic Research Fund, Israel Academy of Sciences. Publication number 430.     

The spectrum problem III: Universal theories

We solve the classification problem and essentially the spectrum problem for universal theories (see [6] for discussion of the meaning of this). We first solve it forT such that ifM ₁,M ₂ elementarily extendM ₀ and are independent over it, then overM ₀ ∪M ₁ there is a prime model. This generalizes [2]. This was subsequently used and generalized for countable first order theories. (This will appear in [5].) But note that there the theory is countable and in the case of structure the model is prime over a non-forking tree of models; here the model is generated by the union (and theT not necessarily countable). The universality is used in Theorey.If T is stable and complete then either (A)for every M ₁<M (l=0, 1, 2)models of T, if M ₀ ⊆M ₁,M ₂, {M ₁,M ₂}is independent over M ₀ (i.e. tp(M ₁,M ₂)is finitely satisfiable in M ₀),then the submodel of M which M ₁ ⋃M ₂ generates is an elementary submodel of M, or (B)there is an unstable theory extending the universal part of T (we can replace universal by Σ₂ and slightly more). Conclusion. For any universalT:Either (a) for every modelM ofT there is a treeI with ≦ω levels and submodelsN _η (η ∈I) of power ≦2^|T| (by [5], just ≦|T|) such that (i)M is generated by ∪_ηεl N _η, (ii)η <v⇒⇒N _η, (iii) ifv is an immediate successor ofη then tp(N _v, ⋃{{N _p:ρ ∈I,v≨ϱ}) is finitely satisfiable inN _η (note that asking this just for quantifier-free formulas is enough).Or (b) for every cardinalλ>|T|, there are 2^γ non-isomorphic models for powerλ. Dedicated to Professor Abraham Robinson The author would like to thank John Baldwin for the interesting talks in September 1980 which led to §3 of this work, Rami Grossberg for various corrections, and the BSF and NSF for their partial support. This paper was originally intended to appear in the Proceedings of the Model Theory Year at the Institute for Advanced Studies, The Hebrew University of Jerusalem, September 1980 — August 1981, published in Isr. J. Math., Vol. 49, Nos. 1–3, 1984.     

Free limits of forcing and more on Aronszajn trees

We prove that the Souslin Hypothesis does not imply “every Aron. (=Aronszajn) tree is special”. For this end we introduce variants of the notion “special Aron. tree”. We also introduce a limit of forcings bigger than the inverse limit, and prove it preserves properness and related notions not less than inverse limit, and the proof is easier in some respects. The result was announced in [9]. The author thanks Uri Avraham for detecting many errors.     

On the existence of large subsets of [λ]κ which contain no unbounded non-stationary subsets RID="★"ID="★" Publication number 698 in author's list. Partially supported by the Israel Science Foundation.

 Here we deal with some problems posed by Matet. The first section deals with the existence of stationary subsets of [λ]^<κ with no unbounded subsets which are not stationary, where, of course, κ is regular uncountable ≤λ. In the second section we deal with the existence of such clubs. The proofs are easy but the result seems to be very surprising. Theorem 1.2 was proved some time ago by Baumgartner (see Theorem 2.3 of [Jo88]) and is presented here for the sake of completeness. Received: 10 December 1998 / Revised version: 2 February 1999 / Published online: 27 March 2002 RID="&starf;" ID="&starf;" Publication number 698 in author's list. Partially supported by the Israel Science Foundation.     

Ultrafilters on -their ideals and their cardinal characteristics

For a free ultrafilter on we study several cardinal characteristics which describe part of the combinatorial structure of . We provide various consistency results; e.g. we show how to force simultaneously many characters and many -characters. We also investigate two ideals on the Baire space naturally related to and calculate cardinal coefficients of these ideals in terms of cardinal characteristics of the underlying ultrafilter.

     

Stable theories

We studyK _T(λ)=sup {|S(A)| : |A|≦λ} and extend some results for totally transcendental theroies to the case of stable theories. We then investigate categoricity of elementary and pseudo-elementary classes. This paper is a part of the author’s doctoral dissertation written at the Hebrew University of Jerusalem, under the kind guidance of Profeossr M. Rabin.     

Lifting problem of the measure algebra

We prove the consistency of “?/I _mz does not split” (see Notation). We write the proof so that with the standard duality, also the consistency of “?/I _tc does not split” (i.e., replacing measure zero by first category, random by generic, etc.) is proved. The method is the oracle chain condition.