# Iterated Forcing and Elementary Embeddings

@inproceedings{Cummings2010IteratedFA, title={Iterated Forcing and Elementary Embeddings}, author={James Cummings}, year={2010} }

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.

## 161 Citations

### A Lifting Argument for the Generalized Grigorieff Forcing

- MathematicsNotre Dame J. Formal Log.
- 2016

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

- Mathematics
- 2013

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

- Environmental Science, Mathematics
- 2021

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

- Mathematics, EconomicsThe 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

- MathematicsThe Journal of Symbolic Logic
- 2009

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

- MathematicsFundamenta Mathematicae
- 2021

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…

### GENERIC LARGE CARDINALS AND SYSTEMS OF FILTERS

- Computer ScienceThe Journal of Symbolic Logic
- 2017

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.

### WEAK SQUARES AND VERY GOOD SCALES

- MathematicsThe Journal of Symbolic Logic
- 2018

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

- Mathematics
- 2011

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

- Mathematics
- 2009

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.

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