A strong polarized relation

Abstract

Abstract. We prove that the strong polarized relationis consistent with ZFC, for a singular µ which is a limit of measurable cardinals.2000 Mathematics Subject Classification. 03E05, 03E55.

Indestructibility of the Tree Property

“…However, we should emphasize that P being λ-liftable does not by itself ensure that any elementary embedding j : V * → M * (for some models V * , M * ) with critical point λ lifts through a P-generic filter G over V * : Typically, it is necessary that j G is an element of M * because then the j(λ)-liftability of j(P) (given by elementarity) ensures that there is a j(P)-generic filter which extends j G, and thus j can lift. 10 See the proof of Theorem 4.7(d) for more details.…”

The ultrafilter number for singular cardinals

“…For a survey of Prikry and Magidor forcing, we suggest the excellent work [5]. So we phrase the theorem in the general setting, but the reader should bare in mind that the full explicit proof for the above theorem is written in [4] just for singulars with countable cofinality.…”

“…We refer to the monograph [6] and the survey papers [1], [3] for general background in this subject. The consistency of the assumptions of the main theorem below is ensured by the forcing construction of [4]. We quote the pertinent result (this is Claim 3.3 of [4]):…”

“…The consistency of the assumptions of the main theorem below is ensured by the forcing construction of [4]. We quote the pertinent result (this is Claim 3.3 of [4]):…”

On the Spectrum of Characters of Ultrafilters

“…The forcing machinery employed in [3] is taken from [4]. The main component of this forcing is an iteration of 'product dominating real', and the last component can be Prikry forcing or Magidor forcing.…”