# Does taking courses in mathematics give any help for mathematical logic?

I'm undergraduate student of philosophy department and I think I'll major in mathematical logic.

For studying mathematical logic, I thought studying math lectures would give help to logic. So I planned to take courses in mathematics department. But I don't have conviction my thought is right. My question is whether taking math courses gives help to mathematical logic or not. Do I have to give up studying those things and just study mathematical logic only?

If taking courses gives any help, I would study same as a math major; that is, I will take analysis, linear algebra, abstract algebra, topology, differential geometry, complex analysis +alpha (maybe real analysis, number theory)ㅡI already learned calculus and set theoryㅡ. If math study is suitable, does any unnecessary course exist among above?

• You could take a course in mathematical logic. Many math departments offer such a course. If no such course is offered the basics of mathematical logic are usually taught in a discrete math course. – Wintermute Jan 9 '15 at 15:43
• for logic, my university math department has 'mathematical logic' only. Are other math courses unnecessary? – fbg Jan 9 '15 at 15:48
• You could take discrete math first which will give you a nice introduction and help you develop some basic proof skills. A lot of math logic courses don't have a prereq, see what your course catalog says. – Wintermute Jan 9 '15 at 16:02
• i omitted that i took logics in discrete math, formal logic , philosophy of mathematics too. – fbg Jan 9 '15 at 16:15

Yes, and no.

It helps because you will see proofs, you will see careful statements and you will learn, even if not directly more examples that can be used later on to study mathematical logic better.

But on the other hand, rarely anyone mentions formal logic in a course about analysis. You don't think about the inference rules, or what sort of statement you're writing, and so on. This might come up in set theory, or in model theory courses, but even then, you might be surprised how little thought we give these processes.

I am teaching a course about naive set theory (well, I'm the TA, but I'm giving a minicourse about extended topics under the guise of an exercise session). In the first class I wrote that if a set is not empty we can pick an arbitrary element of that set.

I've used existential instantiation, to move from $\exists x(x\in X)$ to an actual element $x$ from $X$; of course I didn't mention this. I hoped that they will let it slide, and they did.

But sure enough, when we discussed the axiom of choice, some six weeks later, they asked why can we choose from one non-empty set? How can we be sure that this is doable? And I explained that we have been doing that since the first class. The reason is the rules of logic, which we didn't mention to them -- because it's a naive set theory course, and these rules are applied naively.

To sum up, yes, taking math courses can help (at least those that deal with proofs) and it is very important to learn some basics as well, algebraic structures make excellent examples for theories and models in logic; but don't expect any mention of logic in those courses. You still have to learn it properly on its own.

• Incompleteness is a very long statement using number theory; not much depth in number theory is needed but it would probably help. – Will Jagy Jan 9 '15 at 16:16
• Will, the only result I can recall is the Chinese Reminder Theorem; which you can also treat as a theorem about rings applied to the case of $\bmod n$. And that result is also quite trivial to prove. Last year we gave it as a homework question and I told my students to find it online and copy it. – Asaf Karagila Jan 9 '15 at 16:19
• Fair enough. Can't say I remember much of the details at this point. – Will Jagy Jan 9 '15 at 16:24

Let me tell you how I got interested in logic during my undergrad. In our measure theory class, after constructing Lebesgue measure, Vitali sets were given as examples of Lebesgue non measurable sets. I wondered if one could extend the Lebesgue measure to a measure defined for all sets of reals. My instructor (an expert in applied probability and statistics) showed me how to do one step extension and tried to do a Zorn's lemma argument but he could not handle the limit stages. He forwarded my question to a set theorist who wrote a very interesting answer to me. He first sketched a proof of why continuum hypothesis implies that such a measure does not exist. This I understood without much pain (although, I had no idea then why CH could not be disproved). Then he said that the existence of such measures is equiconsistent with the existence of a measurable cardinal and ZFC is insufficient to prove this consistency. The quoted reason was Godel's incompleteness theorem whose statement I could not understand then. So I took a class in intro logic and suffered through the initial syntax-semantics boring stuff. Unfortunately, I did not find the proof of the incompleteness theorem terribly exciting (I came to a better understanding of incompleteness after learning recursion theory). It wasn't until I took a set theory class in my grad school that I finally learned the techniques needed to understand the proofs of the entire content of that email. I hope your reasons for learning logic turn out to be as exciting as mine did.

Unquestionably, it will help a lot. This is for two main reasons:

1) One needs exposure to proofs and practice solving problems using proofs at a high level. It's much easier to get this at first in dealing with relatively concrete objects (numbers, numerical functions, finite groups, etc.) than with very abstract objects (such as structures in a first-order language).

2) Many of the examples studied in logic come from other areas of math. For example, model theory has major applications to field theory and algebraic geometry. Set theory is interested in the provability of statements coming from all sorts of other areas, often topology or measure theory. Several areas of logic also have significant overlap with theoretical computer science.

You should consider logic a subfield of math, from a practical standpoint. You should be as interested in areas of math outside your field of specialization as non-logician mathematicians would be in areas outside theirs. Abstract algebra (including some amount of linear algebra and number theory) is the first and most important of these you should study. Analysis and topology should also be a priority.

What directions are most important after that will depend on your specialization within logic, and any of the ones you mentioned could conceivably be useful. It is likely that complex analysis and differential geometry would be less important than the others.

Well, mathematical logic is different from what we normally mean when we say 'logic'. To understand where it comes from and some of the motivation, it is generally good to take other math classes, but they will not teach you (much) mathematical logic. Your department should offer classes in logic, and you should take them, but other math classes will show you the context that the problems come from. Without the context, it will be hard for you to understand the mathematical logic. If you specialise in mathematical logic, you will also interact with a lot of logicians that come from a math background, so having taken some basic analysis, algebra, topology and so on will help you communicate with them.

Yes, of course. To master mathematical logic you need to know mathematics, for at least two reasons:

-- methods of mathematical logic are mathematical, and you need to understand how they works from within a core mathematics area of knowledge, tackling pure mathematics issues;

-- mathematical logic developed mostly thanks to mathematicians, if you do not know mathematics to some degree, you are going to miss a real understanding of why and how mathematical logic developed.

I am not sure which math course could be better to attend. I would not go for intro courses of cutting edge issues. Some basic deep understanding of the classic areas is maybe better. Geometry is always important.

Good luck m.