Many authors consider “mathematical logic and set theory” to be one subject, dealing with the study of concepts that are deemed fundamental to math. However, I disagree with this, for a number of reasons I will outline now.
First, categorizing logic and set theory together places undue emphasis on a particular application of logic to set theory, almost implying that this is the “main” application of logic. However, while that may have been true historically, in terms of how logic was first motivated and then applied, now logic is a much more powerful subject that is generally applicable to all sorts of other theories. It is better to separate these two out to emphasize the applicability of each of these subjects on their own.
(As a general follow-up point, regardless of how things may have played out historically, the presentation and organization of subjects can and should evolve over time, especially when there is a good reason for such an evolution.)
Second, I argue in my philosophical post On the Definition of Mathematical Rigor that concepts like numbers and shapes need to taken as fundamental primitives in order to do math. If “logic and set theory” were the subject that studied the concepts fundamental to math, it would then have to include number theory and (elementary) geometry, and this is not an organization that most mathematicians would agree with. Instead, each of logic, set theory, number theory, and geometry study a concept fundamental or primitive to math; they all just study different ones.
Finally, one counter-argument that is likely to be made against the previous paragraph concerns set theory’s ability to formalize the mathematical universe. However, I do not accept the validity of this. No formal system can capture all the theory of an informal primitive, and that includes any axiomatic set theory. Thus, it still wouldn’t make sense to include set theory within the branch that studies the concepts fundamental to math, especially in a privileged position compared to number theory or elementary geometry.
