# Collapsing successors of singulars

@inproceedings{Cummings1997CollapsingSO, title={Collapsing successors of singulars}, author={James Cummings}, year={1997} }

Let κ be a singular cardinal in V , and let W ⊇ V be a model such that κ+V = λ + W for some W -cardinal λ with W |= cf(κ) 6= cf(λ). We apply Shelah’s pcf theory to study this situation, and prove the following results. 1) W is not a κ+-c.c generic extension of V . 2) There is no “good scale for κ” in V , so in particular weak forms of square must fail at κ. 3) If V |= cf(κ) = א0 then V |= “κ is strong limit =⇒ 2κ = κ+”, and also ωκ ∩W ) ωκ ∩ V . 4) If κ = אω then λ ≤ (2א0 )V .

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## References

SHOWING 1-10 OF 18 REFERENCES

### Combinatorial Principle in the Core Model for one Woodin Cardinal

- MathematicsAnn. Pure Appl. Log.
- 1995

### Weak covering without countable closure

- Mathematics
- 1995

The main result of [MiSchSt] is that Theorem 0.1 holds under the additional assumption that card(κ) is countably closed. But often, in applications, countable closure is not available. Theorem 0.1…

### Supercompact cardinals, sets of reals, and weakly homogeneous trees.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1988

It is shown that if there exists a supercompact cardinal then every set of reals, which is an element of L(R), is the projection of a weakly homogeneous tree. As a consequence of this theorem and…

### Chang’s conjecture for ℵω

- Mathematics
- 1990

We establish, starting from some assumptions of the order of magnitude of a huge cardinal, the consistency of (ℵω+1,ℵω)↠(ω1,ω0), as well as of some other transfer properties of the type…

### A Very Weak Square Principle

- MathematicsJ. Symb. Log.
- 1997

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.

### E-mail address: jcumming@andrew.cmu.edu

- E-mail address: jcumming@andrew.cmu.edu