is -complete and , for every . A normal measure over is a fine ultrafilter over with the additional property that every function such that is constant on a set in . Here "constant on a set in " means that there is such that .

Properties

Supercompact cardinals have reflection properties. If a cardinal with some property (say a 3-huge cardinal) that is witnessed by a structure of limited rank exists above a supercompact cardinal , then a cardinal with that property exists below . For

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