An inaccessible cardinal is strongly compact if, and only if, it satisfies thestrong tree property. We prove that if there is a model of ZFC with infinitelymany supercompact cardinals, then there is a model of ZFC where ${\aleph _{\omega + 1}}$ has the strong tree property. Moreover, we prove that everysuccessor of a singular limit of strongly compact cardinals has the strong treeproperty.

If S, T are stationary subsets of a regular uncountable cardinal κ, we say that S reflects fully in T, S < T, if for almost all α ∈ T (except a nonstationary set) S ∩ α stationary in α. This relation is known to be a well-founded partial ordering. We say that a given poset P is realized by the reflection ordering if there is a maximal antichain 〈Xp: p ∈ P〉 of stationary subsets of Reg(κ) so that

We prove that if , and P is an arbitrary well-founded poset of cardinality ≤ κ+ then there is a generic extension where P is realized by the reflection ordering on κ.

A stationary subset S of a regular uncountable cardinal κreflects fully at regular cardinals if for every stationary set T ⊆ κ of higher order consisting of regular cardinals there exists an α Є T such that S ∩ α is a stationary subset of α. Full Reflection states that every stationary set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than κ having the Mitchell order κ++ it is consistent that Full Reflection holds at every λ ≤ κ and κ is measurable.