My economics program had a class that transitions or introduces to proofs with books like Bridge to Abstract Mathematics , Reading, Writing, and Proving: A Closer Look at Mathematics or A Transition to Mathematics with Proofs. But I'm not referring these.

I mean Stephen Cole Kleene's Mathematical Logic like

It begins with an elementary but thorough overview of mathematical logic of first order. The treatment extends beyond a single method of formulating logic to offer instruction in a variety of techniques: model theory (truth tables), Hilbert-type proof theory, and proof theory handled through derived rules. The second part supplements the previously discussed material and introduces some of the newer ideas and the more profound results of twentieth-century logical research. Subsequent chapters explore the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Godel, Church, Turing, and others. The emphasis in the final chapter reverts to logic, with examinations of Godel's completeness theorem, Gentzen's theorem, Skolem's paradox and nonstandard models of arithmetic, and other theorems.

or An Introduction to Mathematical Logic (Dover Books on Mathematics

Topics include the theorems of Gödel, Church, and Tarski on incompleteness, undecidability, and indefinability; a rigorous treatment of recursive functions and recursive relations; computability theory; and Hilbert's Tenth Problem

or Christopher Leary's A Friendly Introduction to Mathematical Logic

In this expansion of Leary's user-friendly 1st edition, readers with no previous study in the field are introduced to the basics of model theory, proof theory, and computability theory. The text is designed to be used either in an upper division undergraduate classroom, or for self study. Updating the 1st Edition's treatment of languages, structures, and deductions, leading to rigorous proofs of Gödel's First and Second Incompleteness Theorems, the expanded 2nd Edition includes a new introduction to incompleteness through computability as well as solutions to selected exercises.

Do economics degrees require this Formal Mathematical Logic. Why?

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    $\begingroup$ "My friends and I don't know any economics degree that requires this Formal Mathematical Logic. Why?" This is something you should probably ask yourself and your friend. My experience is exactly the opposite. I dont know any economics degree that would not require at least surface level covering of majority of the topics covered in the links you provided. $\endgroup$ – 1muflon1 Aug 22 at 22:43
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    $\begingroup$ @1muflon1 i edited question. i don't mean "at least surface level". $\endgroup$ – Nai Aug 22 at 22:44
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    $\begingroup$ I strongly recommend to cover at least some basics of Mathematical Logic. The literature is full with logical flawed articles. Hence, you should master a level to be able to recognize these failures by yourself. $\endgroup$ – Holger I. Meinhardt Aug 23 at 10:02
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    $\begingroup$ The title of your question and the actual question differ. And no, you do not need to learn model theory or proof theory to do economics. $\endgroup$ – Michael Greinecker Aug 23 at 12:31
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    $\begingroup$ @d'Halluin Mathematicians have proved theorems long before these subfields emerged. $\endgroup$ – Michael Greinecker Sep 10 at 7:16

As pointed out in the comments this question asks two different questions. I will try to answer them separately.

Question 1: To what extent does economics use Mathematical Logic?

Answer: while an 'extent' is not something that can be easily measured I think it is fair to say that in theoretical part of economics is virtually omnipresent. Consider the following fact:

MWG Microceonomic Theory - which is arguably the most popular and comprehensive graduate text for microeconomic theory has a rigorous proof derived from rigorous definitions and propositions on virtually every single page (save for preface, practice questions and reference page and few pages here and there). That is about 900 pages out of about 1000 page book that rely heavily on formal mathematical logic.

In different sub-fields of economics the importance of rigorous mathematical proofs can vary but in most of them I cant think of any graduate text that would not lean heavily on formal mathematical logic. Maybe philosophy of economics or history of economic thought would be some exceptions (and even there for example Philosophy of Economics: A Contemporary Introduction by Julian Reiss includes some parts that are written in formal mathematical logic)

Question 2: Do economics degrees require this [referring to subjects you mention inside the question] Formal Mathematical Logic?

Answer: Generally not - I could not find any survey on this issue but I think its fair to say majority of economic degrees do not have such courses that you mentioned as requirement to graduate and obtain a degree. The reason for that is that if you are good at mathematics you will be able to understand and even construct proofs without ever sitting in any course on formal mathematical logic. Any study of mathematics implicitly also teaches you formal mathematical logic - just not in necessary structured or obvious way.

As a matter of fact these fields [referring to the courses you mention] are relatively new subfields of mathematics and as Michael Greinecker pointed out in comments, "Mathematicians have proved theorems long before these subfields emerged." To see this you can just look at the beautiful and rigorous proofs in Euclid's Elements that were drafted thousands of years ago yet most of them still hold up and are quite rigorous.

This being said, taking formal class on introduction into making formal mathematical proofs should improve your own proof-making and understanding of proofs and help you to make your own future work more rigorous. The question is whether this benefit outweighs the opportunity cost of that lost time - but that is opinionated question I won't go into.

I think this can be the best explained by following analogy:

This is like asking to what extent economics uses linguistics and its sub-fields of grammar, stylistics etc. Well again linguistic is omnipresent in economics as most papers are not just proofs with zero text. However, is it necessary to take academic writing class or class in stylistic or grammar to be an economist or to get an economic degree? Well no, I think its fair to say most science degrees will have only optional writing classes and many not even that.

Many people will just learn how to write excellent academic works just by reading other papers. However, at the same time any person's writing would undoubtedly benefit by taking some academic writing class where you are exposed to stylistics and other sub-fields of linguistics even if it is not required to ever have such class.

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