This is in reply to Mike Barr's question about cardinals kappa such that, for all lambda<kappa, the power set of lambda has cardinality <= kappa. (I assume that I've correctly reconstituted the question after Bitnet revised the TeX codes in it.) There are lots (i.e., a closed unbounded class) of such cardinals kappa. To get one (as large as you like), start with any cardinal, iterate exponentiation (with base 2) to produce a countable sequence of larger cardinals, and take the supremum of this sequence. This clearly has the desired property (in fact with < in place of <=). Furthermore, this property is preserved by suprema. So you get lots of such cardinals, without needing any assumptions (like inaccessibles) that go beyond ZFC. Andreas Blass