Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number. A cardinal is called subtle if for every closed and unbounded C and for every sequence A of length for which element number (for an arbitrary ), A there are , , belonging to C, with , such that A =A . A cardinal is called ethereal if for every closed and unbounded C and for every sequence A of length for which element number (for an arbitrary ), A and A has the same cardinal as , there are , , belonging to C, with , such that card( )=card(A A ).