Iterable cardinal

In mathematics, an iterable cardinal is a type of large cardinal introduced by, and , and further studied by. Sharpe and Welch defined a cardinal κ to be iterable if every subset of κ is contained in a weak κ-model M for which there exists an M-ultrafilter on κ which allows for wellfounded iterations by ultrapowers of arbitrary length. Gitman gave a finer notion, where a cardinal κ is defined to be α-iterable if ultrapower iterations only of length α are required to wellfounded. (By standard arguments iterability is equivalent to ω1-iterability.)