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| rdfs:comment | - A Mahlo cardinal (or strongly Mahlo cardinal) is an inaccessible cardinal \(\alpha\) such that the set of inaccessible cardinals below \(\alpha\) is a stationary subset of \(\alpha\) — that is, every closed unbounded set in \(\alpha\) contains an inaccessible cardinal (in which the Von Neumann definition of ordinals is used). The smallest Mahlo cardinal is sometimes called "the" Mahlo cardinal \(M\). (The eponym "Mahlo" has been appropriated as an adjective, so "\(\alpha\) is a Mahlo cardinal" may be rephrased as "\(\alpha\) is Mahlo," for example.)
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| abstract | - A Mahlo cardinal (or strongly Mahlo cardinal) is an inaccessible cardinal \(\alpha\) such that the set of inaccessible cardinals below \(\alpha\) is a stationary subset of \(\alpha\) — that is, every closed unbounded set in \(\alpha\) contains an inaccessible cardinal (in which the Von Neumann definition of ordinals is used). The smallest Mahlo cardinal is sometimes called "the" Mahlo cardinal \(M\). (The eponym "Mahlo" has been appropriated as an adjective, so "\(\alpha\) is a Mahlo cardinal" may be rephrased as "\(\alpha\) is Mahlo," for example.) If we weaken "inaccessible" to merely "regular," we get the weakly Mahlo cardinals. The two definitions are equivalent if the generalized continuum hypothesis is taken to be true. Neither Mahlo nor weakly Mahlo cardinals can be proven to exist in ZFC (assuming it is consistent), not even if we assume the existence of any number of inaccessible cardinals. Nevertheless, it's believed that the existence of these cardinals is consistent with ZFC. The Mahlo cardinals are most relevant to googology through ordinal collapsing functions.
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