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I don't know how to proof a question which I meet in a textbook. It is this: Let $\alpha$ be a successor cardinal, and $S\subset \alpha$ be a statinonary set, then $S$ can be seen as the union of $\alpha$ disjoint stationary sets?

Could someone give me the key points to solve the question?

Any help wil be appreciated. Thanks ahead:)

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1 Answer 1

This fact is called the statonary partition theorem. In the case of successor cardinals, it is due to Ulam, and can be proved using an Ulam matrix. The generalization that every stationary subset of an uncountable regular cardinal $\kappa$ can be partitioned into $\kappa$ many disjoint stationary subsets is due to Solovay, and a proof can found right here in Kanamori's book: The Higher Infinite. An account of the simpler successor case can be found in these slides. There seem to be several proofs on-line, if you google Solovay stationary partition. (It appears that some authors credit Fodor rather than Ulam for the successor case, and so I've become less certain of the history.)

It is interesting to note that the result requires the axiom of choice. Indeed, it is consistent from large cardinals that the club filter on $\omega_1$ is an ultrafilter, in which case one cannot find even two disjoint stationary sets.

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