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Preferably aimed at undergraduates or something an undergraduate can at least understand.

Thank you

J. W. Tanner
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Threnody
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    Halmos - Naive Set Theory is a relatively standard and approachable choice. – Jair Taylor Jul 15 '20 at 04:29
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    Have a look Introduction to Set Theory by Hrbacek and Jech. – Anonymous Jul 15 '20 at 10:50
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    In mathematical practice, you do not need to know about ordinals, and actually all you need to know is the well-ordering theorem. It turns out that there is in fact a short proof of that theorem along the same lines as this short proof of Zorn's lemma: Take any set S. Let F be a choice-function that maps each strict subset A of S to a member of S not in A. We say that (T,<) is a tower iff T is a subset of S and < is a well-ordering of T and ∀x∈T ( x = F(T[<x]) ). Any two towers agree (i.e. one order-embeds into the other). Union all the towers. – user21820 Sep 07 '20 at 15:52

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One of the best books in my opinion on Set Theory for an undergraduate is Classic Set Theory by Derek Goldrei. The author wrote the book particularly for independent study and not just as a teaching guide like many other textbooks so expect it to be different. Some things that I like about it that come up to my mind is how the right margin is filled with helpful notes and that exercises appear as you go along and not necessarily at the end of each chapter with some important exercises solved immediately afterward which helps you maintain the flow of ideas.

Talal Alrawajfeh
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