In mathematics, an iterable cardinal is a type of large cardinal introduced by Gitman (2011), and Sharpe and Welch (2011), and further studied by Gitman and Welch (2011).
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.)
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