I am currently studying Forcing methods in order to understand some independence results and model's constructions.
Now I am interested on formalizing the main notions around forcing such as consistency, completeness, transitive models, well-founded relations, absoluteness, reflection principle, etc. in the logical point of view.
I have heard that Shoenfield has a good work on that, I think I want something related to his approach or something improved (if it uses classical logic).
People said to me that Shoenfield's book Mathematical Logic is a good reference for what I am looking for.
Allerting that I have no trainning in basic logic (but I intent to start it next semester), can someone help me with a study guide with a few reference texts? Which topics should I see in order to have a solid understanding of Forcing in the logical approach?
Thank you
In short, I think Kunen's new book $\textit{Set Theory}$ is probably the best. However I should probably elaborate:
If you want to learn about independence result in set theory, it is a mistake to think that all you need to know is forcing. In general, your goal should be to understand relative consistency.
To understand relative consistency results in set theory, there are several things to understand before you even touch forcing. Relative consistency has logical aspects as well as technical combinatorial aspects.
For the logical prerequisites, you should understand some elementary first order logic including proofs and basic model theory. Then you should know the axiom of set theory and what model of set theory are. You should know of the incompleteness theorems (but knowing the proofs is not really necessary). The incompleteness theorem gives you perspective on why you are studying relative consistency. Next you need to know what consistency means and what relative consistency means.
Then the first major technique is inner model theory. Inner models are just proper class models of set theory. However, proper classes are not sets. There are technical aspects in defining the satisfaction relation for these objects. Here you should understand what relative interpretation means and how consistency result are obtained. Kunen has a nice easy example using $WF$ to prove the consistency of foundation axioms from the consistency of the other axioms. The next very important example is using the constructible $L$ to prove the consistency of $AC$ and $GCH$ from the consistency of the other axioms. Before moving onto forcing, you should understand what it means for $L$ to satisfy $AC$, etc. and how $L$ produces relative consistency result. Understand in general how proper class models are used. Kunen has a nice exposition of this. He defines $L$ using the $L_\alpha$, which is probably easier for beginners than the Jensen $J_\alpha$. (Later if you are interested in determinacy and large cardinals, these type of models become very important.)
Then you can move onto forcing. Kunen using the poset approach. There are some technical aspects involving the forcing language, names, and the truth and definability lemmas. However the consistency of $\neg CH$ is not really all that difficult. However at the beginning, it is more difficult to under the logic behind how forcing produces relative consistency results. Moreover terminology in papers in set theory are somewhat confusing. You may see phrases like "let $M$ be a countable transitive models of ZFC". But it is not proveable in ZFC that there are any model of ZFC! Or you may see phrases like "let $G$ be a generic over $V$". A generic over the whole universe may not exist! There are many other "abbreviations" that come up in more advance papers and text in set theory that make no sense taken literally. Again Kunen is a good place to see logical explanation of how forcing produces consistency results and what the correct mathematics meaning behind common forcing abbreviations. You should take the time to understand the logic and jargon involved in forcing. After understanding this, forcing is really just some combinatorics.
Again in my opinion, the beginner could probably understand line by line all the technical aspect of constructing inner models and forcing extension, but not really understand the main goal of how to use these things to produce independence results. Kunen has the most extension discussion about the logical and meta aspect of inner models and forcing. Essentially all the background to understand independence result can be found in Kunen's $\textit{Set Theory}$, or his $\textit{Foundations of Mathematics}$ which precedes $\textit{Set Theory}$.
Concerning Shoenfield's book, I think it is a bit old. Although I believe Kunen (or possibly Kanamori) mentions that the modern exposition of forcing is due greatly to Shoenfield. Although Chow states that the exposition of forcing is an "open problem", I would disagree. I think the exposition of inner model theory is a more important open problem.