How does first order logic influence everyday mathematics?

Question

I have read through a book about first order logic.

It was interesting. However, when I read through undergraduate math texts it’s unclear about what system they are working in. They don’t specify anything at all about axioms or logical axioms.

I’m so confused now. What is the point of learning about first order logic if it seems that nobody cares / knows very much about it?

Analysis 1 by Tao talks about ZFC and Peano arithmetic but that’s about all I’ve seen.

Is it worth learning model theory? Where do I start with that?

Answer 1

It depends. Set theory, logic, and model theory are their own fields. There are interesting applications of set theory, logic, and model theory to other fields of mathematics, but also plenty of mathematicians work in their own fields without much knowledge of them, only basic knowledge more or less required for all fields of mathematics. In general I think it’s a good idea to know the basics, like what the axioms of ZFC are, what cardinals and ordinals are, what logical quantifiers mean, stuff like that. But anything more than that should be considered as just another field of mathematics. Whether it is worth learning or not depends on your interests and needs.

Answer 2

Most of the time, any particular field of mathematics will be built on some kind of foundation, and the assumption is that the foundation is stable enough for the field to work.

At various times, someone constructs a paradox from some area of mathematics, and there's some work to find a system that avoids that problem. When this happens, some of the theories that rely on the previous system need to be revised since they may no longer be valid. One of the biggest examples of this was in the late 1800s and early 1900s where things like Russell's paradox and Godel's incompleteness theorem revealed issues in set theory and arithmetic which were considered to be some of the most fundamental parts of mathematics.

If you look at this from the view of, say, calculus or probability theory, everything still basically works. It's unlikely you were going to try to integrate over a set that isn't actually a set, or flip a coin based on the completeness of arithmetic. But if you did, then you would now discover that you can't because you're working in a system where those things don't meaningfully exist.

So to answer your question "what's the point of learning about it if no-one cares?" I would say that it's similar to asking "what's the point of learning how to lay a concrete slab?" - if you're an electrician, you probably don't need to know how to do it, but you want to be able to trust that the person who did so knew what they were doing, and maybe it's good to know whether the way the concrete sets will affect the best way to run the cabling along it or something.

In other words, go ahead and learn about it if you find it interesting. Go as deep or as shallow as makes sense for what you're trying to understand. Maybe you'll find the next level of inconsistency in category theory and be the reason everyone has to scramble to fix everything again. Or maybe you'll be like my applied mathematics professor who said something along the lines of "I'm going to take this integral on the assumption that it works, and I'll let the pure mathematicians figure out if I'm wrong".