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I am a year 2 undergraduate student in mathematics, and I have been quite interested in topics like the foundation of mathematics, mathematical logic, set theory, etc. I have studied the basics of calculus, analysis, linear algebra and abstract algebra.

I have a few questions would like to ask:

(a) What could I do to prepare myself in my undergraduate studies if I want to do research / postgraduate studies in these areas? Any courses in undergrad are mostly related? (There is no any course about these topics in my school currently)

(b) It seems like there are not many top graduate schools offering research position on these areas, according to what I searched online. So is it a good choice to do research on this area? Is this an important area in the whole mathematical world?

(c) I read a little bit David Marker's Model Theory: An Introduction, and had quite some difficulties. Is there any easier introductory material on mathematical logic, in particular model theory for me to self-study?

Many thanks!

eurekamath
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    I suggest you talk to someone at your school who both knows you and what the school offers. – Ethan Bolker May 01 '23 at 18:20
  • Have your studies of abstract algebra included an introduction to the structure of mathematical systems, more specifically relations, operations, and functions? – Confutus May 01 '23 at 20:48
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    Kunen has an excellent book, that assuming you have taken some proof based math classes already- is at a perfect level of rigor. I highly reccomend: "The Foundations of Mathematics" by Kunen. It covers an intro to model theory, set theory, and mathematical logic- and is written for undergraduates, in the second half of their degree. – Michael Carey May 01 '23 at 22:50

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First of all, I'm not an expert, so, use my answers with that in mind.

(a) It's difficult to say something about that, because I don't know your courses. But many logic topics use more mainstream maths. For example, Boolean algebras (in first-order logic), some aspects of graph theory (like trees, for set theory), groups (these are recurring examples in model theory)... And, of course, any branch of mathematics serves to motivate questions of logic and the tools are similar (math is math).

(b) Without any doubt, mathematical logic is very important in the world of mathematics. A quick search in zbMATH will allow you to verify that there is a similar number of articles published in the area of logic and in the area of functional analysis, for example. However, it is true that it is not the most common area of work in mathematics. There are important sub-disciplines of logic in which there are currently no more than 100 researchers (for example, I don't think anyone is working exclusively on problems related to infinitary logic). BUT there are researchers who are dedicated to logic but work in a close area. For example, it is common for a model theory researcher to publish papers dealing with questions of algebraic geometry.

(c) If you haven't studied logic before, it might be a bit difficult to start with a model theory book. Many of those introductory books on model theory include sections dealing with questions of first-order logic and set theory, but I would recommend starting with a book that deals specifically with those topics. For example, in this post many examples of introductory texts are given. In addition, many university professors publish on their web pages books of the notes with which they teach their courses. In this link you can find the notes of a course in model theory taught by Thomas Scanlon (I'm not saying that they are good or bad notes, just an example).

Yester
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