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.

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    $\begingroup$ This question will be useful and may answer it for you. I think most of the really important stuff showed up in the answers to your question (some basics on cardinals, ordinals, and well ordering principle/axiom of choice) and I think that can be picked up as you go along. Although, if you want to study set theory or foundations of mathematics then you probably could start now. $\endgroup$ – Paul Plummer Mar 9 '14 at 6:30
  • $\begingroup$ @Paul Plummer: But the answer there by Asaf Karagalia suggests its best to leave axiomatic set theory till the third year. I'm just starting my second. And I'm not taking just Math courses until I transit to a major(hopefully in mathematics) at the end of this year. $\endgroup$ – Ishfaaq Mar 9 '14 at 15:43
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    $\begingroup$ The best time to start studying set theory is when you notice set theory is awesome. $\endgroup$ – Malice Vidrine Mar 9 '14 at 17:05
  • $\begingroup$ Formally constructing the real numbers using only Peano's axioms and the axioms of set theory is a long and difficult process. Then, of course, you have to demonstrate that your construction actually works -- an even more difficult process. If you want to specialize in algebra, number theory or analysis, I wouldn't spend too much time on axiomatic set theory. The foundational issues in these fields were settled long ago. But, by all means, if you are not pressed for time, pick up a good book on set theory for fun. It is something of a hobby of mine. $\endgroup$ – Dan Christensen May 1 '14 at 18:07

Set theory is absolutely necessary to learn more advanced mathematics. It is needed for just about every branch of mathematics, if not every branch. In my opinion, it would be a good idea to start learning some basic set theory notions at least. It will definitely show up in classes like real analysis, complex analysis, and probability just to name a few. Most likely, those undergrad courses will introduce you to some of the theory but to start learning on your own definitely won't hurt. There are numerous resources out there available for learning by yourself. Good luck!

  • $\begingroup$ Yes I think I have the basics quite clear. I even know a bit about cardinality. But I have not tackelled the Axiomatic approach. Just wondering if now would be a suitable moment? $\endgroup$ – Ishfaaq Mar 9 '14 at 9:26
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    $\begingroup$ Sure, no better time than now! $\endgroup$ – homegrown Mar 9 '14 at 15:01

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