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).
Best Answer
One example a bit closer to what you seek might be:
You can see my review, which appeared in the Bulletin of symbolic logic, vol. 11, iss. 2, p. 241, 2005: