Narrow Results

In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2^ω = ω₂ and that satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that □(κ) fails for all regular κ > ω₁.

We extend the results of Laver on using inverse limits to reflect large cardinals of the form, there exists an elementary embedding L_α(V_λ+1) → L_α(V_λ+1). Using these inverse limit reflection embeddings directly and by broadening the collection of U(j)-representable sets, we prove structural results of L(V_λ+1) under the assumption that there exists an elementary embedding j : L(V_λ+1) → L(V_λ+1). As a consequence we show the impossibility of a generalized inverse limit X-reflection result for X ⊆ V_λ+1, thus focusing the study of L(ℝ) generalizations on L(V_λ+1).