# Textbook on elementary set theory that includes all proofs

NB: I have read the earlier post Textbooks on set theory, but the information in that post is not sufficiently specific to answer my question here.

To put it somewhat glibly, I am looking for a book that does for elementary set theory what Edmund Landau's Foundations of Analysis does for analysis.

In other words, I am looking for a book on elementary set theory that explicitly proves everything it asserts, no matter how obvious the assertion or how tedious, or "routine", the proof.

Such a book not only avoids "proofs" such as "obvious", "routine", "exercise" but also the likes of "by induction on $$\alpha$$", or "proof sketches" in general.

As for coverage, the book should at least cover ordinals and cardinals, and, especially, their respective arithmetics.

EDIT: Since this post has received nothing approaching an answer, I think it is in order to relax the requirements somewhat. Please regard the description above as "an ideal to strive for," and propose candidates that you consider approach it most closely.

• You left out some context: why are you interested in a book on elementary set theory that proves all stated results, no matter how obvious?
– KCd
Feb 21, 2022 at 14:33
• The book "How to prove it" by Daniel J. Velleman does this for logic and methods of proof. It covers some set theory. However, it does not lay out axioms in a list as formalism. However, much of set theory is just a rephrasing of sentential logic which is laid out in this book so there's nothing lost here. While if you wanted exactly the axioms of ZFC, you'd need to look in some other reference. Search for "Set theory pdf" on a search engine and you should find quite a few results. Feb 26, 2022 at 18:41

My recommendation is G Takeuti, WM Zaring: Introduction to axiomatic set theory. while they are not proving everything, this book is written as a list of theorems, definitions, claims and of course, lot's of proofs.

as i stated before, there are some proofs that are left to the reader, but the percentage is very low.

right at the beginning of the book the writers are stating what are the language symbols, the logical axioms and rules of inference they are using, and after which in the next few pages you can find statements with one line proof's.

at the beginning of the book they are not skipping any proof of any statement, you will encounter the sentence "the proof is left for the reader" just about in the middle of the book, at this time i was felling pretty confidence in my knowledge to complete those proof's.

if you will read the book, you will find pages filled with consecutive statements and their proof, one after another without an intuition or any explanation beside the proof it self.

may I add that most of the proofs are formal proofs, so there is no non obvious skipping of some steps in the proofs.

hope it helps.

• Thank you! I have not yet had enough time to properly evaluate this book, but I like very much what I've seen of it so far. The authors do seem to take seriously matters such as good notation, unambiguous definitions, and logical sequencing. Conversely, they do not seem to take for granted the reader's foreknowledge of the subject matter (the way that, e.g. Jech does), so I don't expect that they will resort to ambiguous statements and a sloppy exposition.
– kjo
Mar 3, 2022 at 19:34
• I just noticed that, when I hover the cursor over the [+500] (bounty) icon near this answer's upper-left corner, the tooltip pop-up bubble says "This answer has been awarded bounties worth 500 reputation by Community." I am surprised by the "by Community" bit. I thought I had awarded you the bounty. That was definitely intention, anyway. I'm not sure what I did wrong. Be that as it may, I am glad that you got the bounty one way or another. I just wanted to make clear that the end result, however it came about, matches my intentions.
– kjo
Mar 4, 2022 at 14:36
• Hello @kjo, thank you for clarifying your intentions. as mentioned in the last paragraph of of the info page about bouty: "If the bounty was started by the question owner, and the question owner accepts an answer posted during the bounty period, and the bounty expires without an explicit award then we assume the bounty owner liked the answer they accepted and award it the full bounty amount at the time of bounty expiration." so it seems you have done what you intended, but i appreciate your care and most importantly hope you are enjoying the book as much as i did. Mar 18, 2022 at 20:50