Explanation of 1-generic to prove undecidability of halting problem This question is about an answer in question 


  Are there any proofs the undecidability of the halting problem that does not depend on self-referencing or diagonalization ?


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?
A finite prefix of  $G$ is $G \cap \{0,1,\ldots n\}$ for some $n$. It is easier if you look at the characteristic function of $G$ which can we viewed as an infinite word in $2^\omega$. Then a finite prefix is a finite initial part of $G$.

For forcing, there are many resources, depends on what you want to learn. A point to start is the Wikipedia pages and the references listed there: Forcing (set theory), Forcing (recursion theory). Timothy Chow's article A Beginner's Guide to Forcing is a good introduction.