Iterable cardinal

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.)

References

• Gitman, Victoria (2011), "Ramsey-like cardinals I", Journal of Symbolic Logic, 76 (2): 519–540, arXiv:0801.4723, doi:10.2178/jsl/1305810762, MR 2830435, S2CID 16501630
• Gitman, Victoria; Welch, P. D. (2011), "Ramsey-like cardinals II", Journal of Symbolic Logic, 76 (2): 541–560, arXiv:1104.4448, doi:10.2178/jsl/1305810763, MR 2830435, S2CID 2808737
• Sharpe, Ian; Welch, P. D. (2011), "Greatly Erdős Cardinals with some generalizations to the Chang and Ramsey properties", Annals of Pure and Applied Logic, 162 (2): 863–902, doi:10.1016/j.apal.2011.04.002, MR 2817562

External links

• Diagram of iterable cardinals