The tree property at successors of singular cardinals

Assuming some large cardinals, a model of ZFC is obtained in which carries no Aronszajn trees. It is also shown that if is a singular limit of strongly compact cardinals, then carries no Aronszajn trees. Received August 18, 1996     

The tree property at double successors of singular cardinals of uncountable cofinality

Assuming the existence of a strong cardinal κ and a measurable cardinal above it, we force a generic extension in which κ is a singular strong limit cardinal of any given cofinality, and such that the tree property holds at ${\kappa }^{++}$.     

Ordinal definable subsets of singular cardinals

A remarkable result by Shelah states that if κ is a singular strong limit cardinal of uncountable cofinality, then there is a subset x of κ such that HOD_x contains the power set of κ. We develop a version of diagonal extender-based supercompact Prikry forcing, and use it to show that singular cardinals of countable cofinality do not in general have this property, and in fact it is consistent that for some singular strong limit cardinal κ of countable cofinality κ⁺ is supercompact in HOD_x for all x ? κ.     

Souslin trees at successors of regular cardinals

We present a weak sufficient condition for the existence of Souslin trees at successor of regular cardinals. The result is optimal and simultaneously improves an old theorem of Gregory and a more recent theorem of the author.     

The tree property at the first and double successors of a singular

We show that given ω many supercompact cardinals and a weakly compact above them, there is a generic extension in which the tree property holds at the first and second successor of a strong limit singular cardinal.     

Reflecting stationary sets and successors of singular cardinals

REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a which is ⁺ⁿ -supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the bad stationary set. It is shown that supercompactness (and even the failure of PT) implies the existence of non-reflecting stationary sets. E.g., if REF then for many PT(, ₁). In Sect. 2 it is shown that Easton-support iteration of suitable Levy collapses yield a universe with REF if for every singular which is a limit of supercompacts the bad stationary set concentrates on the right cofinalities. In Sect. 3 the use of oracle c.c. (and oracle proper—see [Sh-b, Chap. IV] and [Sh 100, Sect. 4]) is adapted to replacing the diamond by the Laver diamond. Using this, a universe as needed in Sect. 2 is forced, where one starts, and ends, with a universe with a proper class of supercompacts. In Sect. 4 bad sets are handled in ZFC. For a regular {<⁺ : cf<} is good. It is proved in ZFC that if=cf>₁ then {<⁺ : cf<} is the union of sets on which there are squares.     

The tree property at the successor of a singular limit of measurable cardinals

Assume \(\lambda \) is a singular limit of \(\eta \) supercompact cardinals, where \(\eta \le \lambda \) is a limit ordinal. We present two methods for arranging the tree property to hold at \(\lambda ^{+}\) while making \(\lambda ^{+}\) the successor of the limit of the first \(\eta \) measurable cardinals. The first method is then used to get, from the same assumptions, the tree property at \(\aleph _{\eta ^2+1}\) with the failure of SCH at \(\aleph _{\eta ^2}\). This extends results of Neeman and Sinapova. The second method is also used to get the tree property at the successor of an arbitrary singular cardinal, which extends some results of Magidor–Shelah, Neeman and Sinapova.     

Strong tree properties for two successive cardinals

An inaccessible cardinal κ is supercompact when (κ, λ)-ITP holds for all λ?≥ κ. We prove that if there is a model of ZFC with two supercompact cardinals, then there is a model of ZFC where simultaneously ${(\aleph_2, \mu)}$ -ITP and ${(\aleph_3, \mu')}$ -ITP hold, for all ${\mu\geq \aleph_2}$ and ${\mu'\geq \aleph_3}$ .     

A combinatorial property of cardinals

(GCH) For every cardinal there exists such that for every , there are such that .

     

Large cardinals imply that every reasonably definable set of reals is lebesgue measurable

We prove that if there is a supercompact cardinal or much smaller large cardinals, then every set of reals from L(R) is Lebesgue measurable, and similar results. We also introduce some large cardinals. The first author thanks the United States-Israel Binational Science Foundation for partially supporting this research, M. Gitik, J. I. Ihoda and Y. Kopplevich for corrections and Alice Leonhardt for the beautiful typing. Publication no. 241. The second author thanks the National Science Foundation for supporting this research.     

On the free subset property at singular cardinals

We give a proof ofTheorem 1. Let be the smallest cardinal such that the free subset property Fr (, ₁)holds. Assume is singular. Then there is an inner model with ₁ measurable cardinals.     

The cardinals below

The results of this paper concern the effective cardinal structure of the subsets of [ω₁]^<ω₁, the set of all countable subsets of ω₁. The main results include dichotomy theorems and theorems which show that the effective cardinal structure is complicated.     

The strong tree property and weak square

We show that it is consistent, relative to ω many supercompact cardinals, that the super tree property holds at for all but there are weak square and a very good scale at .     

The tree property and the continuum function below

We say that a regular cardinal κ, , has the tree property if there are no κ‐Aronszajn trees; we say that κ has the weak tree property if there are no special κ‐Aronszajn trees. Starting with infinitely many weakly compact cardinals, we show that the tree property at every even cardinal , , is consistent with an arbitrary continuum function below which satisfies , . Next, starting with infinitely many Mahlo cardinals, we show that the weak tree property at every cardinal , , is consistent with an arbitrary continuum function below which satisfies , . Thus the tree property has no provable effect on the continuum function below except for the trivial requirement that the tree property at implies for every infinite κ.     

Better quasi-orders for uncountable cardinals

We generalize the theory of Nash-Williams on well quasi-orders and better quasi-orders and later results to uncountable cardinals. We find that the first cardinal κ for which some natural quasi-orders are κ-well-ordered, is a (specific) mild large cardinal. Such quasi-orders are (the class of orders which are the union of ≦λ scattered orders) ordered by embeddability and the (graph theoretic) trees under embeddings taking edges to edges (rather than to passes). This research was supported by the United States-Israel Binational Science Foundation, grant 1110.