So, I am an absolute beginner in mathematics; only being knowledgeable in some basic ideas of the subject. My interest in math started only recently, while reading about set theory and cardinality (particularly the concept of higher infinities) in some other forums. Can you guys recommend me any farily accessible books or any other material which I could use to understand those topics? Or do I need to study some other areas in mathematics before I am able to comprehend set theory or cardinals?
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2$\begingroup$ I advise to do naive set theory before formal set theory. $\endgroup$– J. De RoCommented Jun 2, 2020 at 21:43
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1$\begingroup$ karagila.org/wp-content/uploads/2016/01/ests-wh.pdf might be helpful. $\endgroup$– Asaf Karagila ♦Commented Jun 2, 2020 at 21:45
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1$\begingroup$ There is a fairly readable introduction by William Weiss, who is a professor at the University of Toronto, that is available on his website: math.toronto.edu/weiss/set_theory.html (you might want to download it since he plans to publish it as a book, at which point the manuscript may or may not continue to be available for free) $\endgroup$– FavstCommented Jun 2, 2020 at 22:04
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1$\begingroup$ I'd recommend checking out Cunningham's Set Theory. It's very self-contained (i.e. no need to worry about missing out on some big prerequisites) and has chapters on both naïve and axiomatic set theory. In the last few chapters he discusses ordinals and cardinals in some detail. Good luck! $\endgroup$– Luke PoeppelCommented Jun 2, 2020 at 22:22
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1$\begingroup$ Halmos - "Naive Set Theory" is quite good. $\endgroup$– Jair TaylorCommented Jun 2, 2020 at 22:41
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3 Answers
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For a real beginner in mathematics who is particularly interested in set theory and cardinalities, I might recommend Stories about Sets by Vilenkin, which is aimed at a high school audience.
I don't recommend studying an axiomatic presentation of set theory until you have significant experience with proofs in one or two other areas of mathematics, such as abstract algebra, analysis, topology or number theory. By an axiomatic presentation, I mean one in which axioms are given for the behavior of sets, such as the "axiom of extensionality" or the "axiom of the power set." This includes the references by Weiss, Halmos and Cunningham mentioned in the comments above. (Strictly speaking, results from other areas of mathematics are mostly not necessary. But there are serious pedagogical and psychological obstacles for a student without any other math background.)
Once you have a sufficient general background in mathematics, Introduction to Set Theory by Hrbacek and Jech is a good choice.
In the meantime, in the course of studying other areas of math, you are exposed to aspects of set theory gradually, with sets presented on an intuitive level. In studying calculus with a modern textbook, you become accustomed to the basic use of set notation. In studying analysis, you learn about countable sets and get practice manipulating sets in more sophisticated ways. For example, Mathematical Analysis by Tom Apostol is an excellent introduction to analysis and has a good (non-axiomatic) chapter on sets.
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If you are an "absolute beginner", then I would recommend starting by working through Book of Proof by Richard Hammack, which builds up basic naive set theory, tours through a variety of good foundations for any math subject, and ends with an introduction to cardinality. There are other similar books you could start with like How to Prove It: A Structured Approach by Velleman and An Introduction to Abstract Mathematics by Bond and Keane, but "Book of Proof" is free.
After gaining a foundation and exposure to mathematical proof in a variety of contexts like that, many introductions to set theory will become accessible.
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I was in a similar boat! I really liked Daniel Cunningham's Set Theory: A First Course (https://www.amazon.com/Set-Theory-Cambridge-Mathematical-Textbooks/dp/1107120322). The text starts very simply with the ZF axioms, then builds up to transfinite recursion, ordinals, and cardinals. I found the last couple chapters challenging, but intriguing.
The book was very readable and interesting, and the proofs are easy to follow. I self-studied the book, having only had limited exposure to formal set theory as a computer science undergraduate.
You probably already realize this, but learning mathematics requires doing as many exercises yourself as possible. I wrote solutions to most of the problems in Cunningham's text, which you can find here: https://sites.google.com/view/dougchartier/mathematics/solutions. Note that I can't vouch for the accuracy of the solutions, but please let me know if you have any corrections or feedback. I know the importance of being able to check your work or get hints if you're stuck (or even just read the solution entirely if you've banged your head against the problem for too long!). I hope they're helpful in your endeavor.
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$\begingroup$ I found your answer interesting. It's the first time I hear of Cunningham's book. Why did you read a book on set theory? Did you pursue further topics in math? $\endgroup$ Commented Aug 31, 2021 at 17:35