I started to learn a few disciplines on my own over the break after my first year in college and one of them was Real Analysis. In the process I came across many issues in Analysis texts concerning the foundations of Mathematics. One issue was the construction of the real numbers and the other being the cardinality of sets. I read Landau's Foundations of Analysis and the first bit of Richard Dedekind's Essays on the Theory of Numbers to console myself about what I thought were naive developments in the first chapters of Analysis texts. For the life of me I still can't explain how numbers are constructed but at least after a first reading of these texts I now know that they can be constructed. And I'm happy.
Now, I was told that the best text for a first glimpse at formal set theory was Naive Set Theory by Paul Halmos. But I found that I was spending too much time on the foundations and the preface on this book hinted it was best to just keep learning your stuff and fill your gaps later so I abandoned Halmos after the first two chapters and have been following Analysis through The Elements Real Analysis by Robert Bartle.
I posted this question today and it made me realise how important a decent footing on set theory actually is. Most of the answers are about axiomatic set theory and although some skimming on google helps me get a slight idea, I do not have a firm grasp on the subject. My question is,
I need some advice as to whether a deviation into learning Set Theory at this point is feasible? When should I, in general, pursue the foundations? Should I at all?
My journey so far in Mathematics is as follows.
I have just started my second year. Have taken courses on first year calculus, multi-variable calculus, Differential Equations, introductory Graph theory, a computational course on matrices, some elementary number theory and a tinge of group theory.
I will be taking another Calculus course, a Linear ALgebra course, another on Differential Equations and one on Linear Programming this semester.
I have also taken a small course on Naive Set Theory and Combinotrics in my first semester. But this just introduced set identities (Associative Laws, Idempotent Laws etc. ) and motivated the use of them in proving relationships between sets. No cardinality or anything beyond.
I am also in the middle of a self-learning project on Analysis and Elementary Number Theory.
Would love some advice. And any would be greatly appreciated.