Coding and Reshaping When There Are No Sharps

Assuming 0 does not exist, κ is an uncountable cardinal and for all cardinals λ with κ ≤ λ < κ+ω, 2 = λ+, we present a “mini-coding” between κ and κ+ω. This allows us to prove that any subset of κ+ω can be coded into a subset, W of κ+ which, further, “reshapes” the interval [κ, κ+), i.e., for all κ < δ < κ+, κ = (card δ)L[W∩δ]. We sketch two applications of this result, assuming 0 does not exist. First, we point out that this shows that any set can be coded by a real, via a set forcing. The second application involves a notion of abstract condensation, due to Woodin. Our methods can be used to show that for any cardinal μ, condensation for μ holds in a generic extension by a set forcing.