6:42 AM
@AlessandroCodenotti I see that your study group chatroom is now close to getting frozen (it is getting close to 14 days of inactivity).
But I suppose that you have most likely already finished reading the text.

6:57 AM
@user3865391 The word "graduate" student is commonly used for PhD degree and also for Masters degree right?
In any case, a PhD student should certainly be comfortable with basic operation with sets, equivalence relations, partial orders.
They should know enough about cardinality, I'd say they should be able to calculate without any problems cardinality of any set they are likely to encounter.
And from beginning PhD student I would also expect to be familiar with Zorn Lemma.

2 hours later…
9:15 AM
BTW Pete L. Clark has some notes on Set Theory on his website: alpha.math.uga.edu/~pete/expositions2012.html
> All the set theory I have ever needed to know. (40 pages total)
> Most of the time, most of us don't need to know more about set theory than the distinction between finite, countably infinite, and uncountable sets. But once in a while it's nice to know a little bit more: e.g. the least uncountable ordinal comes up in topology, or your colleague asks you for a counterexample to the variant of Zorn's Lemma with "chain" replaced by "countable chain." But it is hard to find a treatment of set theory that goes a little beyond Halmos' Naive Set Theory
> or Kaplansky's *Set Theory and Metric Spaces *(both excellent texts) but that isn't off-puttingly foundational and/or axiomatic (i.e., that treats set theory as mathematics, rather than a confusing amalgamation of mathematics and meta-mathematics). For instance, the standard axioms only allow the elements of sets to themselves be sets (so that, e.g., mathematicians do not form a set) and forbid a set from containing itself, although neither of these options seem logically contradictory.
A brief glance suggests that apart from the topics I have mentioned, he also includes ordinals and transfinite induction.

6 hours later…
3:47 PM
There was a recent post on meta, which has a different focus: Where to post for feedback
I'll just quote this part - which seems to be related to foundations and perhaps - at least broadly - to set theory:
> I had difficulty finding a single, concise treatment online of the construction of numbers from the naturals through the reals.
Asking whether there is somewhere a good reference for construction of $\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$ (perhaps also $\mathbb C$) seems like a reasonable question. I have searched a bit - I am surprised that I did not find such question on the main site.
I found this question which has no answers: References for the construction of various number types
This question asks about integers, but the references given there might be interesting also for other number domains.
2

The Number Systems: Foundations of Algebra and Analysis, Solomon Feferman, AMS Chelsea Publishing, 1989 — chapter 4 Retracing Elementary Mathematics, Leon Henkin, W. Norman Smith, Verne J. Varineau and Michael J. Walsh, MacMillan, 1962 — chapter XI Foundations of Analysis, Edmund Landau, AMS Che...

2

Number Systems and the Foundations of Analysis by Mendelson The Number System by Thurston The Structure of Number Systems by Parker

I will also quote two resources mentioned in one of my answers:
> Artmann B: The concept of number (Ellis Horwood, 1988). This books mentions several constructions of reals (Dedekind cuts, Cauchy sequences, decimal representation, continued fractions). Advantages and disadvantages of various approaches are mentioned in this book. ...
> Ethan D. Bloch: The Real Numbers and Real Analysis, Springer, 2001. This book is intended as a textbook for a course in real analysis, but it discusses the two most usual definitions of real numbers in detail in the first two chapters
Does somebody know about other references on this topic? Does somebody know some other suitable references?
I see that one message related to the post on meta mentioned above has been posted also in another chatroom:
in Basic Mathematics, 12 mins ago, by bblohowiak
I had difficulty finding a single, concise treatment online of the construction of numbers from the naturals through the reals. I have collected my notes in a document (<1,200 words); I want feedback so I may correct any errors in my understanding or its expression. Here is a link to my notes; please comment with suggestions: https://docs.google.com/document/d/1_XW4oiZ4jc112jRmWvEuNFUHYTq3SPuxkTkSp7vZuBk/‌​edit