Controlling classical cardinal characteristics while collapsing cardinals

Collapsing the Cardinals of Hod

Assuming that GCH holds and κ is κ+3-supercompact, we construct a generic extension W of V in which κ remains strongly inaccessible and (α+)HOD < α+ for every infinite cardinal α < κ. In particular the rank-initial segment Wκ is a model of ZFC in which (α+)HOD < α+ for every infinite cardinal α.

Large cardinal properties of small cardinals

Forcing Axioms, Supercompact Cardinals, Singular Cardinal Combinatorics

The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtai...

Successor of Singulars: Combinatorics and Not Collapsing Cardinals

On the one hand we deal with (< κ)-supported iterated forcing notions which are (Ê0, Ê1)-complete, have in mind problems on Whitehead groups, uniformizations and the general problem. We deal mainly with the caes 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 b...

- 1 6 Not Collapsing Cardinals ≤ Κ in ( < Κ ) – Support Iterations

We deal with the problem of preserving various versions of completeness in (< κ)–support iterations of forcing notions, generalizing the case " S–complete proper is preserved by CS iterations for a stationary co-stationary S ⊆ ω 1 ". We give applications to Uniformization and the Whitehead problem. In particular, for a strongly inaccessible cardinal κ and a stationary set S ⊆ κ with fat complem...