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 . 

Some consequences of reflection on the approachability ideal

We study the approachability ideal I[κ + ] in the context of large cardinals and properties of the regular cardinals below a singular κ. As a guiding example consider the approachability ideal I(ℵ

Fat sets and saturated ideals

  • J. Krueger
  • Mathematics
    Journal of Symbolic Logic
  • 2003
TLDR
A theorem of Gitik and Shelah is strengthened by showing that if κ is either weakly inaccessible or the successor of a singular cardinal and S is a stationary subset of κ such that NS κ↾S is saturated then κ ∖ S is fat.

Splitting stationary sets in Pκλ for λ with small cofinality

TLDR
A new notion for ideals is introduced, which is a variation of normality of ideals, which proves that there is a stationary set S in Pκλ such that every stationary subset of S can be split into λ many pairwise disjoint stationary subsets.

Distributivity and Minimality in Perfect Tree Forcings for Singular Cardinals

Dobrinen, Hathaway and Prikry studied a forcing Pκ consisting of perfect trees of height λ and width κ where κ is a singular ω-strong limit of cofinality λ. They showed that if κ is singular of

Cofinality changes required for a large set of unapproachable ordinals below ℵ_{+1}

In V, assume that N ω is a strong limit cardinal and 2 Nω = N ω+1 . Let A be the set of approachable ordinals less than N ω+1 . An open question of M. Foreman is whether A can be non-stationary in

FALLEN CARDINALS MENACHEM KOJMAN AND

We prove that for every singular cardinal μ of cofinality ω, the complete Boolean algebra compPμ(μ) contains a complete subalgebra which is isomorphic to the collapse algebra Comp Col(ω1, μ0 ).

Fallen cardinals

Diagonal Prikry extensions

TLDR
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.

The non-compactness of square

TLDR
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.

Exact Upper Bounds and Their Uses in Set Theory

References

SHOWING 1-10 OF 18 REFERENCES

Weak covering without countable closure

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.

  • W. Woodin
  • Mathematics
    Proceedings 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

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

TLDR
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.

Shelah's pcf Theory and Its Applications

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

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