# Large Cardinals and Definable Counterexamples to the Continuum Hypothesis

@article{Foreman1995LargeCA, title={Large Cardinals and Definable Counterexamples to the Continuum Hypothesis}, author={Matthew D. Foreman and Menachem Magidor}, journal={Ann. Pure Appl. Log.}, year={1995}, volume={76}, pages={47-97} }

## 96 Citations

### Approachable Free Subsets and Fine Structure Derived Scales

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- 2021

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…

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- 2008

We present two ways in which the model L(R) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing;…

### Proper forcing and L(ℝ)

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Abstract We present two ways in which the model L(ℝ) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small…

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We show that in the presence of large cardinals proper forcings do not change the theory of L(R) with real and ordinal parameters and do not code any set of ordinals into the reals unless that set…

### PROPER FORCINGS AND ABSOLUTENESS IN L (

- Mathematics
- 2007

We show that in the presence of large cardinals proper forcings do not change the theory of L(R) with real and ordinal parameters and do not code any set of ordinals into the reals unless that set…

### CHANG’S CONJECTURE, GENERIC ELEMENTARY EMBEDDINGS AND INNER MODELS FOR HUGE CARDINALS

- MathematicsThe Bulletin of Symbolic Logic
- 2015

It is proved that a natural principle Strong Chang Reflection implies the existence of an inner model with a huge cardinal and this principle is between a huge and a two huge cardinal in consistency strength.

### An undecidable extension of Morley's theorem on the number of countable models

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We show that Morley’s theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate…

### Absoluteness for the theory of the inner model constructed from finitely many cofinality quantifiers

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- 2021

We prove that the theory of the models constructible using finitely many cofinality quantifiers – C∗ λ1,...,λn and C∗ <λ1,...,<λn for λ1, . . . , λn regular cardinals – is set-forcing absolute under…

### Strongly proper forcing and some problems of Foreman

- MathematicsTransactions of the American Mathematical Society
- 2018

We answer several questions of Foreman, most of which are closely related to Mitchell’s notion of strongly proper forcing. We prove that presaturation of a normal ideal implies projective antichain…

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