What mean $L(\mathbb{R})$ and $L(\mathbb{R})^*$?

Question

I found them relating a cardinality question here.

Does it have anything to do with regularity/computability?

Answer 1

$L(\mathbb{R})$ is the the constructible universe built over $\mathbb{R}$ (i.e. we can use the "real" reals as parameters). Here is the wiki page and here is some information on its construction (beginning on page 3 ~ relative construction). I have never seen $L(\mathbb{R})^*$ before, it also does not appear on the page you have linked to. Then again, I am not a set theorist.