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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?

Hasan Zaeem
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2 Answers2

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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.

David Gao
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    As a side note, most of mathematics is assumed to be carried out in ZFC. That is, if you see a mathematical theorem, it is probably meant as a theorem in ZFC. With that said, not many people will bother mentioning that. After all, mathematics existed long before ZFC is a thing. ZFC certainly puts modern mathematics on a more rigorous footing, but it is by no means necessary. – David Gao Mar 17 '24 at 00:41
  • Do you think I should should approach mathematics and arithmetic the way Euler described it? It wasn’t “rigorous” but he put forth basic definitions of arithmetic and deduced facts using basic logic. – Hasan Zaeem Mar 17 '24 at 01:23
  • @HasanZaeem I don’t think there is such a thing as the “correct” approach to mathematics. How one approaches mathematics highly depends on what they are using it for. That could mean, for example, very different standards of rigor, or different levels of emphasis on calculations, logical deductions, and axiomatic systems. So I can’t really say what you “should” do. What do you find interesting in math? What you want or need mathematics for, now or in the future? The answers to these questions should guide your approach to mathematics, not any overarching philosophy. – David Gao Mar 17 '24 at 01:32
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    Most mathematicians will not be able to state the axioms of ZFC, and it doesn't make a whit of difference. Much more important is to have a basic facility with sets at the level of say Halmos's Naive Set Theory, which is what I think you are driving at. What is needed can be formalized in other ways besides ZFC. – user43208 Mar 17 '24 at 01:49
  • @user43208 That’s a good point. Certainly memorizing the axioms of ZFC are not necessary in any sense. The specific formalism is obviously not the point. (Then again most axioms of ZFC are intuitive enough, so perhaps there isn’t that much of difference if you learn it informally or if you look at the list of axioms once.) – David Gao Mar 17 '24 at 01:57
  • Outside the scope of OP's question, but I'm sure we agree that a professional set theorist will need to be on intimate terms with the details of ZFC. Of particular note are the subtle and powerful ramifications of the Foundation axiom and Replacement Axiom Scheme, but those nuances will not play a role in the lives of most mathematicians beyond routine applications of transfinite induction. – user43208 Mar 17 '24 at 22:24
  • @user43208 That I agree. Most mathematicians don’t need infinite cardinals beyond $2^{2^{\aleph_0}}$ or something of the sort, and you definitely don’t need Replacement for that (then again, I consider Replacement to be quite natural, even if the day-to-day applications of it can all be replaced by other axioms). Foundation seems essentially meaningless if you’re not a set theorist. – David Gao Mar 18 '24 at 01:17
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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".

ConMan
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  • It’s funny because the applied mathematician you reference is like pure mathematicians themselves. “I’ll just assume the foundations are correct, let me construct this parameterisation of a Klein bottle” – Fraser Pye Mar 19 '24 at 03:40