  We define a diffeomorphism invariant of smooth 4-manifolds which we can
estimate for many smoothings of R^4 and other smooth 4-manifolds. Using this
invariant we can show that uncountably many smoothings of R^4 support no Stein
structure. (Gompf has constructed uncountably many smoothings of R^4 which do
support Stein structures.) Other applications of this invariant are given.
