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I am seeking answers from experts in mathematical logic about the amount (if any) of university mathematics I need to know in order to understand mathematical logic and later hopefully do meaningful (independent) research on the subject in general and Godel's Theorems in particular.

I am proficient in high school math and have a bachelor's degree in Physics. I have also recently taught myself some calculus, linear algebra, and parts of real analysis as I assumed you must need at least undergrad math to eventually get proficient in a certain math discipline.

Earlier I had decided to learn up to grad level math but after I glanced through some logic books it appears they make close to zero use of even undergrad math. Also, I have come to know that philosophers too do research in mathematical logic, and as far as I know, they don't study any university math.

So, my question is should I first teach myself undergrad (and grad math) or just dive into mathematical logic as I don't want to later find myself in a position where I have to study all that university math before I can make further progress in logic? If that is the case I would consider enrolling myself in a math program first and doing the research later in the conventional way.

user56417
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  • Do you ultimately want to obtain a college/university position, or is this primarily self-study with the goal of possibly later being able to do some research "on the side" (e.g. as a hobby outside of your "day job")? If the former, then you should go the Masters-Ph.D. route in a graduate department that has some logicians in its faculty. For the Ph.D., just getting past the Ph.D. qualifying exams (U.S. perspective here) is going to require quite a bit of undergraduate and early graduate level math. (continued) – Dave L. Renfro Oct 07 '23 at 14:06
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    If the latter, then I would advise you to immediately dive into logic, and later deal with background issues you might encounter. Possibly of use if self-studying are my off-and-on plans for logic and Peter Smith's excellent guide. There are also many questions in Mathematics Stack Exchange that should be of use if you're self-studying. Regarding physics, there are some areas of overlap between physics and logic (e.g. quantum logic stuff & niche things like this). – Dave L. Renfro Oct 07 '23 at 14:12
  • That is what I am asking. Will I encounter background issues? If yes, I would like to go through the usual routine of college education. If not, then I would just like to do independent research. Like I said I don't later want to find out that I have gaps that need to be filled before I can progress further. – user56417 Oct 07 '23 at 14:17
  • Mathematical logic is pretty narrow and can be safely studied without butting up against the usual stuff in an undergrad degree (hence why philosophers can do it) - but there might come a point where you might see objects from UG math be used as examples... but that isn't going to be a major problem. – Thomas Pluck Oct 07 '23 at 14:19
  • But it would be a minor problem, right? – user56417 Oct 07 '23 at 14:27
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    There's more mathematical flavors where you might need to paper over any gaps in knowledge - but this is a problem for someone who has an advanced foundation in logic, and frankly, its kinda a normal problem for mathematicians themselves. Nobody knows everything and everyone has a gap in their knowledge and detours are normal and necessary.

    Naive set theory or introductory logic isn't something like geometry, where UG math is a non-negotiable prereq.

     – Thomas Pluck Oct 07 '23 at 14:35
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    Thanks, I found this very wise. I think I should just jump into logic then. – user56417 Oct 07 '23 at 14:40
  • I suggest going straight into logic and taking a break to study relevant branches of mathematics whenever it becomes apparent that you need them. Inevitably you will find yourself learning combinatorics, Cantors theory of infinity, computability theory, and various topics of discrete mathematics, but let the domain of logic be your map. I recommend starting with particular formal systems like propositional logic, first order logic, and modal logics. Then, after getting acquainted with the trees, take a step back and examine the forest by studying metalogic. – RyRy the Fly Guy Oct 08 '23 at 01:24

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I would actually disagree, to a certain extent anyways, with the comments. I think that logic is a bit deceptive; there absolutely are some things you should be familiar with before diving into logic, even if they're not routinely used to prove results.

Specifically, I would recommend that anyone interested in logic first master the following topics:

  • Standard arguments by mathematical induction, and in particular $(i)$ the proof of the fundamental theorem of arithmetic and $(ii)$ the construction of a number system where it fails.

  • The $\epsilon$-$\delta$ definition of continuity, the construction of the real numbers via Dedekind cuts and via Cauchy sequences, and the proof that these yield the same mathematical structure.

  • Building off of the previous bulletpoint, basic point-set topology is also fundamental. Specifically, I have in mind comfort with compactness and arguments with open covers.

  • The basics of either group or ring theory, up to the definition of quotients and the proof of the first isomorphism theorem (basically, the part of the subject which is "universal-algebraic").

This is a fair amount of material, essentially what I think of as the first year of a math major at most institutions. The issue is that without this grounding, lots of material in logic will be unmotivated or overly complicated. While it's true that e.g. computability theory - one of the five main branches of logic (the others being set theory, model theory, proof theory, and a catch-all called "nonclassical logics") - can technically be leapt into without any preparation, that's not true of model theory and even for computability theory I think that would result in the student not getting nearly as much out of the subject as they could.

Noah Schweber
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  • Out of curiosity, why the downvote? – Noah Schweber Oct 08 '23 at 01:45
  • As I said, my primary interest is in Godel's Theorems. Are the above topics required there as well? – user56417 Oct 08 '23 at 04:19
  • @user56417 Godel's theorems don't exist in a vacuum; if you want to do "meaningful independent research" on the topic, you do need to be familiar with the milieu they exist in. For instance, the incompleteness theorems are closely related to Godel's completeness theorem, which in turn is a form of compactness (the topological notion!). – Noah Schweber Oct 08 '23 at 04:21
  • Additionally, the whole apparatus of model-theoretic ideas motivating the subject leans a ton on abstract algebraic ideas for motivation; I would expect model theory (the subject the completeness theorem lives in) to be almost impenetrable to someone not familiar with a bit of abstract algebra. The construction of the reals is the least directly relevant component, but it's such a pivotal moment in mathematics for illustrating the potential utility of sets that if you ever want to look at set theory at all you'll need it. – Noah Schweber Oct 08 '23 at 04:23
  • Will this topological notion not be explained in logic books? Or even if lack of knowledge of it makes life harder but not impossible even then I would like to skip. – user56417 Oct 08 '23 at 04:24
  • @user56417 It will be explained somewhat, but - since they're logic books, not topology books - in my experience generally not well enough for someone with no prior experience in topology to follow everything. – Noah Schweber Oct 08 '23 at 04:25
  • Can I study it later on when I need it? Or will it require a long detour? – user56417 Oct 08 '23 at 04:30
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    I think you are taking a wrong approach. Assuming a minimalist " Only study what I need" approach is not ideal. If you want to do research, you will be solving new problems with no known solution method. You will need to reference the vast collection of known proof techniques to gain intuition on how to solve some arbitrary problem. It's not about learning about some definition or terminology, it's about learning how mathematics gets done, across a wide range of concepts, because if it was known how to solve the problem; it wouldnt be a research problem. – Michael Carey Oct 08 '23 at 19:39
  • agreed, not the best approach – RyRy the Fly Guy Oct 10 '23 at 04:04