Let $L^{{10}^{100}}$ be the sublanguage of an arithmetic language $L$ consisting of formulas of $L$ with ${{10}^{100}}$ symbols or fewer. is there a Turing Machine M that when run on an empty input eventually outputs every true sentence of $L^{{10}^{100}}$ without ever outputting a false sentence of $L^{{10}^{100}}$? If yes, how does it work?
I think the answer is yes and the machine would run over different states and each state outputs a symbol that encodes a true sentence of $L^{{10}^{100}}$. Any ideas?
Sure, but it's not very interesting. There are only finitely many true sentences under the required length limit. So there is a computer program that consists of hard-coding exactly these sentences and then printing them out.
This is an example of the well-known but sometimes confusing fact that every finite set is computable (you're only asking for c.e., but any computable set is c.e.).