A note on canonical functions

We construct a generic extension in which the ℵ₂nd canonical function on ℵ₁ exists. Supported by NSF and by a Fulbright grant. Publ. 378. Partially supported by the B.S.F.     

Can you feel the double jump?

When p = c/n and c goes from less than one to greater than one, the random graph G(n, p) experiences the double jump. The first order language is too weak to recognize this change while there are properties expressable in the second order monadic language for which the change is clear. © 1994 John Wiley & Sons, Inc.     

Uniformization and the diversity of Whitehead groups

Techniques of uniformization are used to prove that it is not consistent that the Whitehead groups of cardinality ℵ₁ are exactly the strongly ℵ₁-free groups. Some consequences of the assumption that every strongly ℵ₁-free group of cardinality ℵ₁ is Whitehead are derived. Other results about uniformization are also proved. Research partially supported by NSERC grant #9848. Research partially supported by the BSF. The authors thank Rutgers University for its support. Publication #441.     

Number of open sets for a topology with a countable basis

LetT be the family of open subsets of a topological space (not necessarily Hausdorff or evenT ₀). We prove that ifT has a countable base and is not countable, thenT has cardinality at least continuum. Partially supported by the Basic Research Fund, Israeli Academy of Sciences. Publication no. 454 done 6,8/1991. Partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

How much sweetness is there in the universe?

We continue investigations of forcing notions with strong ccc properties introducing new methods of building sweet forcing notions. We also show that quotients of topologically sweet forcing notions over Cohen reals are topologically sweet while the quotients over random reals do not have to be such. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On certain indestructibility of strong cardinals and a question of Hajnal

A model in which strongness of is indestructible under ⁺ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of { |2 <}.     

On reconstructing separable reducedp-groups with a given socle

Let \(\bar B^* \) be a separable reduced (abelian)p-group which is torsion complete. We ask whether for \(G \subseteq \bar B^* \) there is \(H \subseteq _{pr} \bar B^* ,H[p] = G[p]\) ,H[p]=G[p],H not isomorphic toG. IfG is the sum of cyclic groups or is torsion complete, the answer is easily no. For otherG, we prove that the answer is yes assuming G.C.H. Even without G.C.H. the answer is yes if the density character ofG is equal to Min _n<ω|p ⁿG|, i.e., $$\mathop {Min}\limits_{n< \omega } |p^n G| = \mathop {Min}\limits_m \mathop \Sigma \limits_{n > m} |(p^n G)[p]/(p^{n + 1} G)[p]|$$ Of course, instead of two non-isomorphic we can get many, but we do not deal much with this.     

How large can a hereditarily separable or hereditarily Lindelöf space be?

The main result of this paper is that ifV satisfies GCH andω ₁<λ<μ are arbitrary regular cardinals, then in some cardinal preserving forcing extensionW ofV we have λ^<λ=λ=2 ^N 0,μ=2 ^λ and there are a hereditarily separableX ?2 ^λ with |X|= \(2^{\aleph _0 } \) =μ and a hereditarily LindelöfY ?2 ^μ withw(Y)= \(2^{2^\aleph 0} \) =μ. So far similar results have only been obtained under the assumption of CH.     

Dominating Functions and Graphs

A graph is called dominating if its vertices can be labelledwith integers in such a way that for every function : thegraph contains a ray whose sequence of labels eventually exceeds. We obtain a characterization of these graphs by producinga small family of dominating graphs with the property that everydominating graph must contain some member of the family.     

The two-cardinals transfer property and resurrection of supercompactness

We show that the transfer property for singular does not imply (even) the existence of a non-reflecting stationary subset of . The result assumes the consistency of ZFC with the existence of infinitely many supercompact cardinals. We employ a technique of ``resurrection of supercompactness'. Our forcing extension destroys the supercompactness of some cardinals; to show that in the extended model they still carry some of their compactness properties (such as reflection of stationary sets), we show that their supercompactness can be resurrected via a tame forcing extension.