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I could not see $|H(\kappa)| > 2^{<\kappa}$.

It is a question in Kunen book. The other part is answered, this part may be clear but I could not see.

Also, $2^{<\kappa}$ is not clear for me. Is it a set of functions which from $\kappa$ to 2?

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    You probably mean $\geq$ rather than $>$ since the latter is not true.2011-05-23
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    @Apostolos: See my answer for that comment.2011-05-23

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