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I’m interested in the foundations of mathematics. I just completed the book First Order Mathematical Logic by Angelo Margaris. Next planning to study the book The Foundations of Mathematics followed by Set Theory, both by Kenneth Kunen.

While it is usually said that set theory has no prerequisites, I’ve seen set theory books often giving examples from Topology and Analysis. Since I want to follow these examples, I would like to know the following:

  1. What are the key concepts from Topology and Analysis a set theorist should know?
  2. What are the recommended textbooks (or online materials) that cover these concepts? Please suggest books that cover the absolute minimum and not full-fledged books on these topics.

Thanks in advance!

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  • $\begingroup$ There's a difference between studying set theory for its own sake and in studying it as a foundation for other branches of mathematics. I don't need to know about natural numbers to understand set union, but I do if I want to understand the significance of $x \cup \{x\}$ as defining the successor function on natural numbers. $\endgroup$
    – chepner
    Oct 23 at 22:28
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I don't think you need much topology or analysis at all.

It is however very difficult to work through an advanced text on axiomatic set theory, like Kunen's Set Theory, without having the mathematical maturity of at least an advanced undergraduate student. So, without experience with mathematical rigour (like you'd usually learn in a first course on Topology, Analysis, Group Theory, Measure Theory, and so on), it may be hard to appreciate the subtleties of set theory (and set theory is filled to the brim with subtleties).

If you've never worked through a basic text on Analysis or on Topology prior to learning Set Theory, then I'd recommend doing that just for the sake of becoming mathematically mature.


I'm not aware of books only covering the absolute minimum in Topology or Analysis, since the minimum necessary for Set Theory is too little to write a book about. In general, any undergraduate introduction to Topology or Analysis will suffice, but here are some specific references:

Topology

The most advanced topology needed is likely product topologies and compactness.

  • Chapter 2 from Munkres' Topology will likely cover about 95% of the topology you need to understand Kunen's text.
  • These notes by Allen Hatcher cover enough. And you likely won't need chapter 4.

Analysis

As far as Analysis goes, if you know what a Cauchy sequence is, you're probably covered on that front as well. You can certainly ignore anything about power series or complex numbers.

  • I recommend Terence Tao's Analysis I for someone interested in Set Theory. Up to Chapter 5 should be plenty for a bare minimum, but I'd say there's useful extra information in chapters 6 to 9 as well.
  • The first four chapters of Rudin's Principles of Mathematical Analysis.
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  • $\begingroup$ the minimum necessary for Set Theory is too little to write a book about --- My feeling is that it might be better to revise this to: "the minimum necessary for Set Theory is [of] too little [general interest] to write a book about". $\endgroup$ Oct 23 at 18:18

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