I'm doing my Bachelor's Thesis on the continuum in $L$ and $L[0^\sharp]$.

In $L$ I study the gaps without new reals (sets of natural numbers) in the hierarchy, as presented in Gaps in the constructible universe, Marek and Srebrny, 1973. Of course, the issue is deeply linked to the fine structure of $L$, as studied in The fine structure of the constructible hierarchy, Jensen, 1972. I've also looked into some pathological properties of the continuum in $L$, such as the existence of a $∆_2^1$ nonmeasurable set.

I haven't found much literature on any of these two topics regarding $L[0^\sharp]$. Certainly, many of the gap results are easily generalizable to $L[a]$, but I'd like to know whether anyone has studied further how the information codified in $0^\sharp$ might affect the hierarchy, or spawn benign or pathological properties.

I have found some talk of $L[0^\sharp]$ in Friedman's Fine structure and class forcing, but they don't dwell on these details and the book is also highly advanced.

Thanks in advance :)



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