Generalized Notion
There is yet a third notion, model theoretic in nature and sometimes referred to as generalized stationarity. This notion is probably due to Magidor, Foreman and Shelah and has also been used prominently by Woodin.
Now let be a nonempty set. A set is club (closed and unbounded) if and only if there is a function such that . Here, is the collection of finite subsets of .
is stationary in if and only if it meets every club subset of .
To see the connection with model theory, notice that if is a structure with universe in a countable language and is a Skolem function for, then a stationary must contain an elementary substructure of . In fact, is stationary if and only if for any such structure there is an elementary substructure of that belongs to .
Read more about this topic: Stationary Set
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