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In the book "Set Theory" of Thomas Jech, it is defined for any two stationary subsets $S,T$ of a regular uncountable cardinal $\kappa$, $S<T$ if and only if $S\cap\alpha$ is stationary in $\alpha$ for almost all $\alpha\in T$.

The author previously states:

"In the context of closed unbounded and stationary sets we use the phrase for almost all $\alpha\in S$ to mean that the set of all contrary $\alpha\in S$ is nonstationary".

I don't know what the author means by "contrary", thanks for any help.

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1 Answer 1

up vote 2 down vote accepted

If $P$ is some statement about ordinals, the statement

$P(\alpha)$ is true for almost all $\alpha\in S$

means that $\{\alpha\in S:P(\alpha)\text{ is false}\}$ is non-stationary. By the set of all contrary $\alpha$ he means simply $\{\alpha\in S:P(\alpha)\text{ is false}\}$.

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Thank you for your answer –  Camilo Arosemena Jan 7 '13 at 17:44
    
@Camilo: You’re welcome. –  Brian M. Scott Jan 7 '13 at 17:46

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