TITLE: "Introduction to mathematical logic" book from a formalist perspective
QUESTION [8 upvotes]: I'm looking for books that introduce the reader to mathematical logic assuming the perspective of a formalist. 
I've found that many books are more or less written for the platonist - like Kunen's Foundations of Mathematics, where he even implicitly says on pp. 191 that his book, if I understood it right, is primarily written for platonists, but also explains how a formalist would understand his book (in a whopping 3 pages compared to a couple of sentences for platonist view!).
I'm looking for a book that doesn't wait until page 191 to explain this to me, but constantly conveys the formalist viewpoint.
It is important that the books clearly explains the distinction between theory and metatheory and where different theorems of the metatheory live in (e.g. the soundness theorem can be perceived to be a theorem of ZFC since the relevant parts of the metatheory can be coded in ZFC).
I looked at every book from the thread Ask for recommendations for textbook on mathematical logic and none was what I was looking for. Closest to my needs came Kunen - who at least mentions formalism and how his book should be read according to this perspective. This contrasts with other logic books who don't mention anything, and Cori and Lascar's book - for their excellent introduction concerning the vicious circle in what mathematical logic studies - and Goldrei's book on logic, which is not on the list.
To give an explanation for this, perhaps, unusual request: I find it that I understand mathematical theories best when the setting in which the theory "lives" in is clearly outlined so that working in that theory is just formal manipulations of symbols - of course I can attach meaning and intuition to these manipulations, but there has to be a "fixed" setting to work in. From what I've read this aligns understanding aligns best with the perspective of formalism. But sadly mathematical logic is always somewhat vague and in basic core always seems to be somewhat obscure (Kunen says in the above mentioned book for example on page 190 that 

we cannot say exactly

what metatheory is. Now I accept that we can't begin with formal setting based on nothing, because there has to be an informal description of the most basic formal elements of our setting, but I would hope that there are books that explain in more detail that in a single paragraph what metatheory really is. Additionally the lecturer at a course I'm taking also believes in some absolute mathematical objects - I assume he is a platonist - since he frequently says things like "no, now we're not talking about a formalized version of the natural numbers, we're talking about the real natural numbers", which totally annoys me because for me, there are no real natural numbers).

REPLY [7 votes]: I'd like to elaborate a bit on the suggestion, mentioned by some others, that your complaint is not really about platonism versus formalism, but about the common practice of being not completely clear about the metatheory.  Given that the distinction between theory and metatheory is so important, why don't the books make it completely clear at all times which metatheory is in force?
To some extent this can be blamed on poor exposition, but I'd like to suggest that there are good reasons behind this practice, which are important to understand.  As an example, consider the irrationality of √2.  The proof of this theorem is almost always presented in textbooks without any explicit statement about what formal system the proof is supposed to be taking place in.  So how is the reader supposed to tell whether the proof is correct?  Wouldn't it be better to state at the beginning that the proof is supposed to be carried out on the basis of 1st order Peano arithmetic, or whatever?
Well, this could be done, but there are a couple of reasons why this is not typically done.  The most important is that almost everyone finds it easier to understand the proof that √2 is irrational if it is presented in the usual manner.  Even if they want to verify a statement such as "`√2 is irrational' is a theorem of PA", they find it easier to do this by first getting an intuitive understanding of the proof, and then making a "second pass" through the proof to verify that each step in the argument can be mimicked by a formal deduction in PA, rather than taking a purely formal string such as ∀a∀b¬(Sa⋅Sa=SS0⋅Sb⋅Sb) and mechanically checking each step of a formal proof of it.  A second reason is that it is often the case that we want to verify that "√2 is irrational" is a theorem not only of PA, but of various other formal systems.  Again, while one possible approach is to go through the entire formal verification process every time one wants to switch to another formal system, it is far more practical if the reader can understand the "content" of the argument and verify for himself or herself that all the necessary steps can be carried out on the basis of whatever formal system is currently of interest.  The flexibility is useful.
My guess is that despite your stated views, you count yourself among the vast majority of people who are able to read and verify the correctness of the proof that √2 is irrational without having to have it completely formalized first.  After all, almost everyone who studies mathematics encounters this argument before learning the details about formal systems.  My guess is that the trouble begins when the theorems in question get a lot more complicated.  For example, consider the proof of Goedel's 1st incompleteness theorem.  This is a lot more complicated than the proof that √2 is irrational, and there might be parts that you have difficulty with.  This is the point where you might worry about the validity of what's being asserted, and might wish that the metatheory were clearly defined so that you could fall back on formally verifying that A follows from B on the basis of the metatheory.  I'd like to suggest, though, that almost always, it's not the lack of clarity about the metatheory that's the real problem; it's that the reasoning is just not being explained clearly enough for you.  With rare exceptions, if the reasoning is correct, it's going to be correct in any "reasonable" choice of metatheory.
At this point you might say, "I understand everything you're saying, but still, the textbooks and professors sometimes talk about sets and integers as if they're real things, and I just don't believe that they're real!  So I can't follow what they're arguing, and I have a sneaking suspicion that they're using philosophically unjustified assumptions about the reality of integers or sets in their supposedly mathematically rigorous proofs."  Don't worry, this isn't what is happening.  Any time that someone refers to the "actual" integers or the "standard model" or something like that, you can always make sense of it by taking your metatheory to be some set theory such as ZFC, inside which there is a unique set of natural numbers.  People don't always say ZFC explicitly, for the same reason as before: It doesn't have to be ZFC; various other set theories would work just fine, and it's useful to be flexible.
In the end, I think that rather than look for a book that presents things from a formalist point of view, you are better off developing the skill of reading a proof and recognizing what assumptions are needed to carry it through.  Eventually you need to develop this skill anyway.  For example, for the proof of Goedel's 2nd incompleteness theorem, you need to be able to go through the proof of the 1st incompleteness theorem and verify that it can be formalized in (for example) PA.  If you have this skill then you don't need to be explicitly told which metatheory is in force.  And probably the best way to develop the skill is to do some formalization exercises.  Looking for such exercises (perhaps in the context of modern proof assistants such as Coq or Mizar) might be more fruitful than looking for a formalist textbook.