For a regular uncountable cardinal κ and a cardinal λ with cf(λ) < κ < λ, we investigate the consistency strength of the existence of a stationary set in ${}_{\kappa }\lambda$ which cannot be split into λ⁺ many pairwise disjoint stationary subsets. To do this, we introduce a new notion for ideals, which is a variation of normality of ideals. We also prove that there is a stationary set S in ${}_{\kappa }\lambda$ such that every stationary subset of S can be split into λ⁺ many pairwise disjoint stationary subsets.

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional “inner model like” properties. In particular, in this model, the class of Mahlo cardinals reflecting stationary sets is the same as the class of weakly compact cardinals, and every regular Jónsson cardinal is weakly compact. On the other hand, we force and construct a model for the level by level equivalence between strong compactness and supercompactness...

Working in L[E], we examine which large cardinal properties of κ imply that all stationary subsets of cof(<κ) ∩ κ⁺ reflect.

We prove two theorems concerning strong compactness, measurability, and the class of supercompact cardinals. We begin by showing, relative to the appropriate hypotheses, that it is consistent non-trivially for every supercompact cardinal to be the limit of (non-supercompact) strongly compact cardinals. We then show, relative to the existence of a non-trivial (proper or improper) class of supercompact cardinals, that it is possible to have a model with the same class of supercompact cardinals in...

In §1 we define some properties of ideals by using games. These properties strengthen precipitousness. We call these stronger ideals. In §2 we show some limitations on the existence of such ideals over ${}_{\kappa }\lambda$. We also present a consistency result concerning the existence of such ideals over ${}_{\kappa }\lambda$. In §3 we show that such ideals satisfy stronger normality. We show a cardinal arithmetical consequence of the existence of strongly normal ideals. In § 4 we study some “large cardinal-like” consequences of stronger...

The relations M(κ,λ,μ) → B [resp. B(σ)] meaning that if $A\subset {\left[\kappa \right]}^{\lambda }$ with |A|=κ is μ-almost disjoint then A has property B [resp. has a σ-transversal] had been introduced and studied under GCH in [EH]. Our two main results here say the following: Assume GCH and let ϱ be any regular cardinal with a supercompact [resp. 2-huge] cardinal above ϱ. Then there is a ϱ-closed forcing P such that, in $V^P$, we have both GCH and $M({\varrho }^{(+\varrho +1)},{\varrho }^{+},\varrho )\nrightarrow B$ [resp. $M({\varrho }^{(+\varrho +1)},\lambda ,\varrho )\nrightarrow B\left({\varrho }^{+}\right)$ for all $\lambda \le {\varrho }^{(+\varrho +1)}]$. These show that, consistently, the results of [EH] are sharp. The necessity...

Let κ < λ be regular cardinals. We say that an embedding j: V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V. Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ⁺⁺-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ⁺⁺-tall measurable cardinal κ. Now more generally, suppose that κ ≤ λ are regular and one wishes the GCH to fail at λ with κ being λ-supercompact. Silver’s methods show...

We force and construct a model containing supercompact cardinals in which, for any measurable cardinal δ and any ordinal α below the least beth fixed point above δ, if ${\delta }^{+\alpha }$ is regular, δ is ${\delta }^{+\alpha }$ strongly compact iff δ is δ + α + 1 strong, except possibly if δ is a limit of cardinals γ which are ${\delta }^{+\alpha }$ strongly compact. The choice of the least beth fixed point above δ as our bound on α is arbitrary, and other bounds are possible.