I once decided that I want to understand real analysis. It's looks exciting, construction of sets of numbers and something else, I forgot already. I learned some logic, propositional logic and predicate logic(the one with quantifiers, functions, relations and etc-etc). But it seems I did something wrong. I still cannot understand that set theory I need to learn in order to understand real analysis. These axioms are too confusing, I mean their symbolic representation. No one really explains how those axioms sound. When I compare axiom in English and axiom in Logical Symbols, I get confused. They do not like even nearly the same. For example, if I try to translate to you "28 ударов ножом", I say "28 stab wounds". It has this meaning. But if you really try look at russian expression you'll be confused, you'll say "Stop there's nothing about wounds and stab wounds...". Yes, it is. If to translate literally to English, it says "28 punches by a knife". Sure it sounds somewhat weird, but it still conveys proper meaning. And what happens when I read some texts on axiomatic set theory, I compare the two expressions that are actually written in different languages(logic and English), I get confused.

They just left expressions without literal translations. But translation is not the main thing, I can do it myself for the most part. The problems is that meanings of literal and proper translations are still seem to be very different. They do not explain why they convey the same meaning. For example I can explain to you why is "28 punches by a knife" is the same as "28 stab wounds".

And another problem is that most of the time they do not say for what variables stand for, like variables y, x, w... They give you an expression and they say what it means in english without actually mentioning what variables correspond to what things(I mean, sets, elements of the sets and so on.)

When these three problems merge together, it nearly impossible to understand anything. It really like trying to understand some asian language with their weird looking symbols.

I need your help. What have I do to learn real analysis? May be I have to continue to learn logic? May be that's why I do not automatically understand what axioms are trying to say? Is that because I am not enough "mathematically tough"? I don't even know mathematical analysis. Or the problem is not only in me, may be it is the texts I read so far aren't so much good?

Where did I might have messed up? Can you tell me how you learned set theory and real analysis? What books or textbooks or anything else were helpful to you? Thank you. Yo will really help me and anyone else who struggles with the same type of problem, if there are such people.


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    $\begingroup$ You may find this intro course helpful. I did it a few years ago.. Professor Francis's style of teaching is awesome for starters... $\endgroup$ – rsadhvika Aug 7 at 10:30
  • $\begingroup$ @Tim Solnze which real analysis text(s) have you tried studying? $\endgroup$ – erdoswiles Aug 7 at 13:14
  • $\begingroup$ Axiomatic Set Theory, by Suppes, is a short clear intro to the basics. I just googled it to get the exact title and found to my surprise that it is presently available as a FREE pdf. $\endgroup$ – DanielWainfleet Aug 7 at 15:46

I would recommend Tom Apostol's Calculus. It starts with an introduction to basic set theory and then develops the properties of the real numbers. Then it moves into one-variable real analysis at a relatively relaxed pace but is nonetheless completely rigorous. I would suggest you study the first few chapters till you become comfortable with the basic notions such as limits and handling inequalities, and then transition to a more standard book like Rudin or Abbott. Note that this book treats topics in an unusual order with integration being developed before differentiation and sequence/series are treated in detail only later. I don't see this as much of a disadvantage because you are currently wanting to only get a transition into standard real analysis. Try to do at least a few exercises, especially at the beginning, from each chapter.


I followed a course on set theory in my third year of studying mathematics. The formal logic language, though useful and very important for lying the foundation of mathematics is also hard to intuitively get, and I would recommend it as a place to start learning mathematics.

Instead I would start with calculus and some basic set theory, like the intro course rsadhvika recommended. Play around with some of the rigorous but more intuitive definitions there. If you want to understand real analysis topology might also be very worth your while after getting to grips with calculus.

Logic and axiomatic set theory, though a very interesting mathematical subject are not necessary to understand real analysis, so you can skip those altogether if you want. Personally I would revisit them after some time, though doing some propositional logic first might help with understanding the axiom systems.

  • $\begingroup$ Okay, thank you. I will try :) $\endgroup$ – Tim Solnze Aug 7 at 11:32
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    $\begingroup$ Did you mean you would not recommend set theory as a place to start learning math? You wrote that you would recommend it, but seemed to argue against it $\endgroup$ – J. W. Tanner Aug 7 at 22:10
  • $\begingroup$ I would recommend some basic set theory, i.e. set operations, how to construct $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$ and perhaps a little bit about cardinals. Axiomatic set theory, which it seems the OP has started with I would not recommend as a starting place to get into mathematics given the very formal language used in logic, though I found it to be very intersting so I would recommend it at a later time after you have gotten used to mathematical rigour. $\endgroup$ – Floris Claassens Aug 8 at 9:20
  • $\begingroup$ @J.W.Tanner: In my opinion, one should not attempt to learn ZFC set theory when learning real analysis. But one must understand first-order logic (FOL), preferably many-sorted FOL, so that one can then write down axioms for real numbers and just the little bit of set theory needed for real analysis, such as the ability to construct a set of reals satisfying some FOL property. Real analysis is about proving theorems from those axioms, and not about the foundational endeavour to prove that a structure satisfying those axioms exist (which is very important but is not quite real analysis). $\endgroup$ – user21820 Aug 21 at 18:04

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