# Canonical structure in the universe of set theory: part two

@article{Cummings2006CanonicalSI, title={Canonical structure in the universe of set theory: part two}, author={James Cummings and Matthew D. Foreman and Menachem Magidor}, journal={Ann. Pure Appl. Log.}, year={2006}, volume={142}, pages={55-75} }

## 34 Citations

### Diagonal Prikry extensions

- MathematicsThe Journal of Symbolic Logic
- 2010

In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X then there is an L-like inner model Kx which Y covers V”, which always include GCH and Global Square.

### Covering Matrices, Squares, Scales, and Stationary Reflection

- Mathematics
- 2014

In this thesis, we present a number of results in set theory, particularly in the areas of forcing, large cardinals, and combinatorial set theory. Chapter 2 concerns covering matrices, combinatorial…

### ON TODD EISWORTH ’ S CHAPTER FOR THE HANDBOOK OF SET THEORY : “ SUCCESSORS OF SINGULAR CARDINALS ”

- Mathematics
- 2016

This chapter offers a comprehensive and lucid exposition of the questions and techniques involved in the study of combinatorics of successors of singular cardinals. What is so special about…

### KNASTER AND FRIENDS III: SUBADDITIVE COLORINGS

- MathematicsThe Journal of Symbolic Logic
- 2022

We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals θ < κ, the existence of a…

### SINGULAR CARDINAL COMBINATORICS

- Mathematics
- 2004

From May 1 to May 6, 2004 24 set theorists met at the Banff International Research Station to discuss Singular Cardinal Combinatorics. Descriptions of the contents of their talks will be published in…

### Lower consistency bounds for mutual stationarity with divergent cofinalities and limited covering

- Computer Science, Mathematics
- 2019

This work greatly reduces the reliance on covering properties in the proof of consistency strength of mutually stationary sequences of sets concentrating on points with divergent cofinality, and shows that if $\kappa$ is a J\'onsson cardinal with $kappa < \aleph_\kappa then the sharp for a model with a strong cardinal exists.

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It is shown that supercompactness (and even the failure of PT) implies the existence of non-reflecting stationary sets, and that under suitable assumptions it is consistent that REF and there is a κ which is κ+n-supercompact.

### The non-compactness of square

- MathematicsJournal of Symbolic Logic
- 2003

This note proves that in the resulting model every stationary subset of ℵω+1 ⋂ cof(ω) reflects to an ordinal of cofinality ω1, that is to say it has stationary intersection with such a ordinal.

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We consider saturation properties of ideals in models obtained by forcing with countable chain condition partial orderings. As sample results, we mention the following. If M[G] is obtained from a…

### Saturated Filters at Successors of Singulars, Weak Reflection and Yet Another Weak Club Principle

- MathematicsAnn. Pure Appl. Log.
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A very weak version of the principle □ discovered by Jensen who proved it holds in the constructible universe L , which is strong enough to include many of the known applications of □, but weak enough that it is consistent with the existence of very large cardinals.

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We show that the construction of an almost free nonfree Abelian group can be pushed from a regular cardinal /C to ~IC+I. Hence there are unboundedly many almost free nonfree Abelian groups below the…

### Squares, scales and stationary Reflection

- MathematicsJ. Math. Log.
- 2001

Interactions between these three theories in the context of singular cardinals are considered, focusing on the various implications between square and scales (a fundamental notion in PCF theory), and on consistency results between relatively strong forms of square and stationary set reflection.

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. We show that the transfer property ( @ 1 ; @ 0 ) ! ( (cid:21) + ;(cid:21) ) for singular (cid:21) does not imply (even) the existence of a non-reﬂecting stationary subset of (cid:21) + . The result…

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Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary…