# Thread: Set theory question.

1. ## Set theory question.

Hi all,

In Kunen's book, one of the exercises is the following:

Show that for $\displaystyle \kappa > \omega$, |H($\displaystyle \kappa$)| =$\displaystyle 2^{< \kappa}$.

But, H($\displaystyle \omega$) = V($\displaystyle \omega$) and |V($\displaystyle \omega$)| = $\displaystyle \omega$. Also, $\displaystyle 2^{< \omega}= \omega$ (because $\displaystyle 2^{< \omega}$ is the cardinality of the union of omega many finite sets (the characteristic functions for each n)).

Is this wrong?

Thanks in advance!

Sam

2. The notation used in this post is so non-standard, I think that you need to define what each symbol means or how it is used.

3. Ok, so:

$\displaystyle \kappa$ is used as a variable over cardinals.
|$\displaystyle x$| denotes the cardinality of $\displaystyle x$.
$\displaystyle \alpha^{<\beta}$ is the cardinality of the union of $\displaystyle \alpha^{\delta}$ for $\displaystyle \delta < \beta$.
H($\displaystyle \kappa$) denotes the set of sets which are hereditarily of cardinality $\displaystyle < \kappa$.
V($\displaystyle \alpha$) is the $\displaystyle \alpha^{th}$ level in the cumulative hierarchy.

Hope this helps.

Please say if I've left anything important out.

Regards
Sam

4. Just a tangential point: the notation used is not non-standard as far as I am aware. Kunen's book is a standard text, which contains all of this notation except V(\alpha). In Kunen R is used instead of V. In most modern treatments, though, V is used. See Von Neumann universe - Wikipedia, the free encyclopedia, for an example.

Regards
Sam