Iterated Forcing and Elementary Embeddings

  title={Iterated Forcing and Elementary Embeddings},
  author={James Cummings},
I give a survey of some forcing techniques which are useful in the study of large cardinals and elementary embeddings. The main theme is the problem of extending a (possibly generic) elementary embedding of the universe to a larger domain, which is typically a generic extension of the ground model by some iterated forcing construction. 

A Lifting Argument for the Generalized Grigorieff Forcing

Another class of forcing notions which preserve measurability of a large cardinal k from the optimal hypothesis, while adding new unbounded subsets to k is described, which admits fusion-type arguments which allow for a uniform lifting argument.

Group radicals and strongly compact cardinals

We answer some natural questions about group radicals and torsion classes, which involve the existence of measurable cardinals, by constructing, relative to the existence of a supercompact cardinal,

The Variety of Projection of a Tree-Prikry Forcing

We study which κ-distributive forcing notions of size κ can be embedded into tree Prikry forcing notions with κ-complete ultrafilters under various large cardinal assumptions. An alternative

Ramsey-like cardinals

  • V. Gitman
  • Mathematics, Economics
    The Journal of Symbolic Logic
  • 2011
New large cardinal axioms generalizing the Ramsey elementary embeddings characterization are introduced and it is shown that they form a natural hierarchy between weakly compact cardinals and measurable cardinals.

Large cardinals and definable well-orders on the universe

A reverse Easton forcing iteration is used to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals, by choosing the cardinals at which coding occurs sufficiently sparsely.

Characterizing large cardinals through Neeman's pure side condition forcing

We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $\omega_2$ in forcing extensions by the pure side


The notion of ${\cal C}$ -system of filters is introduced, generalizing the standard definitions of both extenders and towers of normal ideals and investigating the topic of definability of generic large cardinals properties.


A model with weak square but no very good scale at a particular cardinal, starting from countably many supercompact cardinals, where □K, is produced.

The Proper Forcing Axiom

The Proper Forcing Axiom is a powerful extension of the Baire Category Theo- rem which has proved highly effective in settling mathematical statements which are independent of ZFC. In contrast to the

Forcing when there are large cardinals: an introduction

Two ideas in the context of large cardinal forcing are described, which may not be familiar to those who have not worked in this area, that uses large cardinals to obtain interesting properties for singular cardinals.



Identity crises and strong compactness

Abstract. From a proper class of supercompact cardinals, we force and obtain a model in which the proper classes of strongly compact and strong cardinals precisely coincide. In this model, it is the

A weak generalization of MA to higher cardinals

We generalized MA e.g., to ℵ1-complete forcing, by strengthening the ℵ2-C.C. condition which occurs in many proofs. We show some consequences of MA generalized, and show that we get a model of ZFC in

Notes on Singular Cardinal Combinatorics

We present a survey of combinatorial set theory relevant to the study of singular cardinals and their successors. The topics covered include diamonds, squares, club guessing, forcing axioms, and PCF

On certain indestructibility of strong cardinals and a question of Hajnal

A model in which strongness ofκ is indestructible under κ+ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number

The Tree Property

Abstract We construct a model in which there are no ℵn-Aronszajn trees for any finiten⩾2, starting from a model with infinitely many supercompact cardinals. We also construct a model in which there

Martin's Maximum, saturated ideals and non-regular ultrafilters. Part II

We prove, assuming the existence of a huge cardinal, the consistency of fully non-regular ultrafilters on the successor of any regular cardinal. We also construct ultrafilters with ultraproducts of

On the singular cardinals problem I

We show how to get a model of set theory in which ℵω is a strong limit cardinal which violates the generalized continuum hypothesis. Generalizations to other cardinals are also given.

Set theory - an introduction to independence proofs

  • K. Kunen
  • Mathematics
    Studies in logic and the foundations of mathematics
  • 1983
The Foundations of Set Theory and Infinitary Combinatorics are presented, followed by a discussion of easy Consistency Proofs and Defining Definability.

Making the supercompactness of κ indestructible under κ-directed closed forcing

A model is found in which there is a supercompact cardinal κ which remains supercompact in any κ-directed closed forcing extension.