We study the consequences of stationary and semi-stationary set reflection. We show that the semi-stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of the weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss, and prove that they follow from stationary and semi-stationary set reflection augmented with a weak form of Martin’s Axiom. We also show that there are some differences between the two reflection principles, which suggests that stationary set reflection is analogous to supercompactness, whereas semi-stationary set reflection is analogous to strong compactness.

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.