I have a working knowledge of calculus and linear algebra. But when I pick up books on mathematical logic (for example the ones listed in the logic study guide by Peter Smith), they often use mathematics I am not familiar with. Is it possible for a non-mathematician to gain an in-depth understanding of mathematical logic in general and Godel's theorems in particular or should I first familiarise myself with undergraduate level mathematics?
While mathematical logic is mathematical (it's right there in the name), as a principled targeted study, there is not much overlap with the usual mathematical undergraduate curriculum (at least in the US).
That is to say, knowing calculus and analysis and geometry and topology will not help substantively with mathematical logic except with the very general skill of manipulating symbols well.
Or more simply, you do not -need- a UG degree in mathematics to understand basic mathematical logic mostly because mathematical logic won't be covered there (but it couldn't hurt! and the math curriculum is infused with logical thinking that mathematical logic would seem almost trivial obvious by the end). What you'll get with a math education is the tolerance for a lot of greek letters (and realizing 'oh you have to substitute the definition for the greek letter and move symbols around and you're almost done').
A good place to get an introduction to logic is in a computer science discrete math class (usually a very early class in that curriculum, so it is very accessible). Another academic source is a class in doing proofs (usually a prereq for getting into the math major). This is often called a 'bridge' class because it helps people who have a STEM background 'cross over' into mathematical maturity, meaning they have a good idea what it means to 'know' something in mathematics, what you have to 'get' things mathematically. And usually part of this class is ... and/or/not/ifthen/forall/thereexist.
Looking at Peter Smith's Logic: A Study Guide I can see now why you might have trouble. All those 'Introduction to First Order Logic' And 'Introduction to Hemiunibal Trigalactic Intercalumniate Spaces of size Zero' are ... clears throat ... not for the beginner and not an introduction. It's like you have to know the subject first before reading them.
You may want to get a text that is more like 'Beginner's Guide to Logic'. Trust me those are not the simplest things ever either but you won't be barraged with new symbols right in the forward to the preface. If you learn basic truth tables and things like modus ponens and DeMorgan's laws and manipulating 'for all' and 'there exist' you're practically there.
And Daniel Velleman's short text 'How to Prove It' or Daniel Solow's book, 'How to Read and Do Proofs: An Introduction to Mathematical Thought' are both -very- accessible and will give you some basic logic operators to work with.
If you are trying to come at Gödel's theorem from a philosopher's point of view without lots of crazy symbolics (you gotta have some symbols - it -is- math) there are a few ways.
Nagel and Newman's "Gödel's Proof" is an excellent and short intro, but may be a bit fast, and while accessible, the new vocab may be dense.
Hofstatder's Gödel, Escher, Bach is what I'd call a popularization of Nagel and Newman's book. It is -very- lengthy, but is entertaining the whole way through. You'll learn a hundred things a long the way (including propositional and predicate logic) and a better understanding of the vocab in Nagel and Newman.
As to 'in-depth'... how do I put this... Suppose I want an in-depth understanding of how black holes work. Do I need an undergrad education in physics (or similar STEM)? Yeah I kinda think you do and then a few years of grad study in exactly black holes, doing all the calculations, reading all the papers. That's what 'in-depth' means.
But can you get a -good- grasp of black holes... um Gödel's theorem with only a philosophy background as long as you get a grounding in basic mathematical logic (propositional and predicate logic).
Ya know, it's not a bad life skill to be able manipulate and/or/if/not/there exists/for all, so even a Schaum's outline on Logic will really go a long way.
Introductory books on logic vary in the amount of symbolic logic they expect the reader to be able to cope with. Though in any undergraduate level text, even one aimed at philosophy or liberal arts students, you should expect to find some formal logic. As a minimum, an introductory text is going to teach you propositional logic and quantifier logic, and without those you will not be able to advance as far as Gödel's theorems.
On the whole, my advice is just get stuck in and learn the symbolic logic used in textbooks like Peter Smith's "Introduction to Formal Logic". It may be unfamiliar, but it is not all that hard if you persevere with it.
If you find Peter Smith a little heavy going, a good online and freely downloadable introduction to logic can be found in the forallx project at https://forallx.openlogicproject.org/
If you want something with a more philosophical angle, containing some symbolic logic but with a fairly light approach, an interesting alternative is John MacFarlane's "Philosophical Logic: A Contemporary Introduction".
If you would like something rather more meaty that covers both philosophical and mathematical aspects of logic, my current favourite introductory text is Harrie de Swart's "Philosophical and Mathematical Logic" which is part of the Springer Undergraduate Texts in Philosophy series.
In my opinion, I do not think that having undergraduate maths background is necessary to understand the mathematical logic materials. For me, I only have basic maths background but I can still learn mathematical logic by myself.
About my experience in modern textbook
As a self learner, I have read few modern introductory textbooks, like A concise introduction to logic by Patrick J. Hurley. The edition that I read contains few errors, for example, in the section about the definite description which was proposed by Bertrand Russell, the definition given by book does not fulfill the uniqueness condition, using a conditional sign instead of a biconditional sign.
For other similar modern elementary level textbooks, I think some of them are not well designed in terms of pedagogy. Those textbooks often start with the translation between symbolic language and natural language, then suddenly a leap to introducing some metalogic results and the given sketch of proof often appeal to rules or metatheorems that are not given or shown with derivations. For more advanced modern textbooks, I found that sometimes they emphasis the distinction between metalanguage and object language in the text but not in their actual practices. For this reason, sometimes I have to guess whether the new introduced symbols belong to the part of metalanguage or the object language.
The suggested classic textbooks
For elementary level, I would suggest
- Introduction to logic and to the methodology of deductive sciences (by A. Tarski but not the J. Tarski one)
The above book includes a less formal treatment but basic concepts for beginners in mathematical logic with daily mathematics as illustrative examples.
- Mathematical Logic (by S.C. Kleene)
This book seems introduced a standard treatment of the modern development of logic, like using model theoretical semantics with daily mathematics as illustrative examples.
- Introduction to Symbolic Logic and its Application (by Rudolf Carnap)
This book introduced the basic treatment of mathematical logic and showed how the meaning of the more advanced mathematical concepts, like isomorphism, structure can be captured with symbolic language.
- Methods of Logic (By W.V.O Quine)
For more advanced topics (like about incompleteness theorem),
- Introduction to Metamathematics (S.C. Kleene)
- Introduction to Mathematical Logic (A. Church)
- Logical Syntax of Language (R. Carnap)
- Mathematical Logic (W.V.O Quine) This is more related to the Russell's Principia but last chapter with protocol sentences
I am here to share my personal experience as a self learner in mathematical logic. This is not a study guide given by the experts in this field. I am just suggesting you to take a quick look on these books and find the suitable one for yourself. Enjoy your study journey :)
A brief look at Peter Smiths guide shows such textbooks as Goldreis set theory, Cutlands intro recursion, etc.
The mathematical prerequistes for many of these books is quite minimal. So long as you have some experience with proof-based work you will be fine. In fact, many comp sci and philosophy programs (in the US) will attempt a basic logic course terminating either in Godels first incompleteness or applications of compactness, so it is quite possible to learn without an undergraduate degree in math.
In fact, most undergrad degrees in math will barely touch on mathematical logic. Such is the stated point of Smith's guide in the first place, I think.