Mahlo cardinal: Difference between revisions
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First, you can see that any weakly Mahlo cardinal is regular. Assume \( \kappa \) is weakly Mahlo but not regular. Let \( \lambda_i \) be a sequence of cardinals with limit \( \kappa \) and length \(\eta < \kappa \). Let \( C^* = \{\lambda_i+1: i < \eta\} \) and let \( C \) be the closure of \( C^* \). You can verify that \( C^* \) is unbounded, and thus \( C \) is club, however \( C \) doesn't contain any regular cardinals. Contradiction! Similarly, you can show that any weakly Mahlo cardinal is a limit cardinal. Assume \( \kappa = \lambda^+ \) for some \( \lambda \). Let \( C = \{\lambda + \eta: 0 < \eta < \kappa \} \). Then \( C \) is club, however \( C \) doesn't contain any regular cardinals. Contradiction! You can continue on to show that the least inaccessible cardinal isn't weakly Mahlo, and any weakly Mahlo cardinal has to be a limit of weakly inaccessibles: if \( \kappa \) is the least weakly inaccessible, then the set of limit cardinals below \( \kappa \) is club but doesn't contain any regulars, and similarly if \( \kappa \) is the next weakly inaccessible after \( \lambda \), then the set of limit cardinals in-between \( \lambda \) and \( \kappa \) is club but doesn't contain any regulars.
Continuing on this way, a weakly Mahlo cardinal can be shown to be weakly hyper-inaccessible. There's a convenient characterisation of weakly Mahlo which explains why they're so large. Recall that [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] is the least ordinal \( \alpha > \omega \) so that, for any \( \Delta_1(L_\alpha) \)-definable function \( f: \alpha \to \alpha \), there is a closure point of \( f \) below \( \alpha \). By weakening the definability condition, you get
Weakly Mahlos see some proof-theoretical usage in ordinal-analysis of extensions of Kripke-Platek set theory, such as KPM, since, like how \( \Omega \) acts as a "diagonalizer" over the Veblen hierarchy, warranting its use in OCFs, a Mahlo cardinal can be thought to act as a "diagonalizer" over the inaccessible hierarchy.
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