This question is about an answer in question
Bjørn Kjos-Hanssen answer states that:
You can first show that the halting problem (the set $0'$) can be used to compute a set $G\subseteq\mathbb N$ that is 1-generic meaning that in a sense each $\Sigma^0_1$ fact about $G$ is decided by a finite prefix of $G$. Then it is easy to prove that such a set $G$ cannot be computable (i.e., decidable).
I have trouble understanding this sentence as I am new to forcing. For example, what does "finite prefix" mean here? Can someone explain this sentence? (I know what $\Sigma^0_1$ means.)
Can someone explain the general idea behind forcing? I am also looking for easy to read introductions articles and books about forcing.
edit: So, how can we prove that each $\Sigma_1^0$ fact about $G$ being decided by a finite prefix leads to the fact that $G$ cannot be computable?