# Set theory question.

• Dec 23rd 2010, 02:05 AM
sroberts
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?

Sam
• Dec 23rd 2010, 04:36 AM
Plato
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.
• Dec 23rd 2010, 04:56 AM
sroberts
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
• Dec 23rd 2010, 05:01 AM
sroberts
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