  We study the approachability ideal I[\kappa^+] in the context of large
cardinals properties of the regular cardinals below a singular \kappa. As a
guiding example consider the approachability ideal I[\aleph_{\omega+1}]
assuming that \aleph_\omega is strong limit. In this case we obtain that club
many points in \aleph_{\omega+1} of cofinality \aleph_n for some n>1 are
approachable assuming the joint reflection of countable families of stationary
subsets of \aleph_n. This reflection principle holds under Martin's maximum for
all n>1 and for each n>1 is equiconsistent with \aleph_n being weakly compact
in L. This characterizes the structure of the approachability ideal
I[\aleph_{\omega+1}] in models of Martin's maximum.
