# Large cardinals and definable well-orders on the universe

@article{BrookeTaylor2009LargeCA, title={Large cardinals and definable well-orders on the universe}, author={Andrew D. Brooke-Taylor}, journal={The Journal of Symbolic Logic}, year={2009}, volume={74}, pages={641 - 654} }

Abstract We use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle , at a proper class of cardinals κ. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.

## 22 Citations

### Large cardinals and locally defined well-orders of the universe

- MathematicsAnn. Pure Appl. Log.
- 2009

### LARGE CARDINALS AND LIGHTFACE DEFINABLE WELL-ORDERS, WITHOUT THE GCH

- MathematicsThe Journal of Symbolic Logic
- 2015

A class forcing is constructed that adds a well-order at every inaccessible cardinal and preserves ZFC, all cofinalities, the continuum function, and all supercompact cardinals and can apply the forcing to answer a question about the definable failure of the GCH at a measurable cardinal.

### Coding into HOD via normal measures with some applications

- Mathematics, ChemistryMath. Log. Q.
- 2011

We develop a new method for coding sets while preserving GCH in the presence of large cardinals, particularly supercompact cardinals. We will use the number of normal measures carried by a measurable…

### HOD, V AND THE GCH

- Mathematics, EconomicsThe Journal of Symbolic Logic
- 2017

Starting from large cardinals, a model of ZFC is constructed in which the GCH fails everywhere, but such that GCH holds in its HOD.

### Strong combinatorial principles and level by level equivalence

- MathematicsBollettino dell'Unione Matematica Italiana
- 2022

We construct via forcing models for the level by level equivalence between strong compactness and supercompactness in which δ holds on a stationary subset A of the least supercompact cardinal. We may…

### More on the preservation of large cardinals under class forcing

- Mathematics
- 2018

We introduce first the large-cardinal notion of $\Sigma_n$-super\-com\-pact\-ness as a higher-level analog of the well-known Magidor's characterization of supercompact cardinals, and show that a…

### Subcompact cardinals, squares, and stationary reflection

- Mathematics
- 2011

We generalise Jensen’s result on the incompatibility of subcompactness with □. We show that α+-subcompactness of some cardinal less than or equal to α precludes $${\square _\alpha }$$, but also that…

### Inner models with large cardinal features usually obtained by forcing

- MathematicsArch. Math. Log.
- 2012

A variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals are constructed, and three general proof methods are described, which can be used to prove many similar results.

### Indestructibility of Vopěnka’s Principle

- MathematicsArch. Math. Log.
- 2011

Vopěnka’s Principle and VopĚnka cardinals are shown to be indestructible under a broad class of forcing constructions, specifically, reverse Easton iterations of increasingly directed closed partial orders.

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