Prikry-Type Forcings

  title={Prikry-Type Forcings},
  author={Moti Gitik},
One of the central topics of set theory since Cantor has been the study of the power function κ→2 κ . The basic problem is to determine all the possible values of 2 κ for a cardinal κ. Paul Cohen proved the independence of CH and invented the method of forcing. Easton building on Cohen’s results showed that the function κ→2 κ for regular κ can behave in any prescribed way consistent with the Zermelo-Konig inequality, which entails cf (2 κ )>κ. This reduces the study to singular cardinals. 

Ordinal definable subsets of singular cardinals

A remarkable result by Shelah states that if κ is a singular strong limit cardinal of uncountable cofinality, then there is a subset x of κ such that HODx contains the power set of κ. We develop a

Prikry-type forcings after collapsing a huge cardinal

. Some models of combinatorial principles have been obtained by collapsing a huge cardinal in the case of the successors of regular cardinals. For example, saturated ideals, Chang’s conjecture,

Prikry-type forcing and minimal α-degree

In this paper, we introduce several classes of Prikry-type forcing notions, two of which are used to produce minimal generic extensions, and the third is applied in α-recursion theory to produce

Approachable Free Subsets and Fine Structure Derived Scales

Shelah showed that the existence of free subsets over internally approachable subalgebras follows from the failure of the PCF conjecture on intervals of regular cardinals. We show that a stronger

Two-cardinal ideal operators and indescribability

. A well-known version of Rowbottom’s theorem for supercompactness ultrafilters leads naturally to notions of two-cardinal Ramseyness and corresponding normal ideals introduced herein. Generalizing

Silver type theorems for collapses

  • M. Gitik
  • Mathematics
    Ann. Pure Appl. Log.
  • 2020

Prikry-type forcing and the set of possible cofinalities

It is known that the set of possible cofinalities pcf(A) has good properties if A is a progressive interval of regular cardinals. In this paper, we give an interval of regular cardinals A such that

Appalachian Set Theory 2006–2012: Short extender forcing

In order to present this result, we approach it by proving some preliminary theorems about different forcings which capture the main ideas in a simpler setting. For the entirety of the notes we work


In order to present this result, we approach it by proving some preliminary theorems about different forcings which capture the main ideas in a simpler setting. For the entirety of the notes we work


We survey applications of ultrafilters and ultrafilter constructions in two set theoretic contexts. In the first setting, that of large cardinals, we explore a number of large cardinal properties and



Surveys in set theory

1. Iterated Forcing James E. Baumgartner 2. The Yorkshireman's guide to proper forcing Keith J. Devlin 3. The singular cardinals problem independence results Sharon Shelah 4. Trees, norms and scales

Blowing up power of a lingual cardinal - wider gaps

  • M. Gitik
  • Mathematics
    Ann. Pure Appl. Log.
  • 2002

A measurable cardinal with a closed unbounded set of inaccessibles from ()

We prove that o(κ) = κ is sufficient to construct a model V [C] in which κ is measurable and C is a closed and unbounded subset of κ containing only inaccessible cardinals of V . Gitik proved that

On gaps under GCH type assumptions

  • M. Gitik
  • Mathematics
    Ann. Pure Appl. Log.
  • 2003

Indiscernible Sequences for Extenders, and the Singular Cardinal Hypothesis

Extender based forcings

It is shown that it is consistent to have “ c f κ = ℵ 0 . GCH below κ, 2 κ > κ + , and ” and answers the question from Gitik's On measurable cardinals violating the continuum hypothesis.

A power function with a fixed finite gap everywhere

An application of the extender based Radin forcing to cardinal arithmetic to construct a model satisfying 2κ = κ+n together with 2λ = λ+n for each cardinal λ < κ.

On closed unbounded sets consisting of former regulars

  • M. Gitik
  • Mathematics
    Journal of Symbolic Logic
  • 1999
A method of iteration of Prikry type forcing notions as well as a forcing for adding clubs is presented and it is shown that the strength of above is at least ο ( κ ) = κ .

On the Singular Cardinals Problem

In this paper we show, for example, that if the GCH holds for every cardinal less than tc, a singular cardinal of uncountable cofinality, then the GCH holds at tc itself. This result is contrary to