I'm a brazilian math undergraduate student. I study at UFAM (Universidade Federal do Amazonas). My scientific initiation advisor is from the area of Functional Analysis. However, he also wanted to learn about Forcing (so maybe he can apply this technique to his work in the future). We then started to study Set Theory from Kunen's book "The Foundations of Mathematics" (TFM). We are thinking about finishing it first and only then study the book "Set Theory" (ST) and begin this one by already skipping the first chapter (which will be already covered by TFM). By this moment, we are nearly close to finish the first chapter of TFM and begin to study the second chapter (which talks about models and formal proofs). However, my research is supposed to end next year in december (along with my undergrad). So I wanted to know if you think the plan to get to study Forcing in these one and a half years is realistic or if it's not enough time. Also I would like to know if there are some things I could not study in TFM now in order to study sooner the ST one. Thanks in advance!
2 Answers
It pretty much depends on your background in logic, but I think that it is fairly reasonable to start directly with Kunen's Set Theory and check The Foundations of Mathematics whenever it is needed in order to save time, if you're worried about it. Happily, in the former Kunen cites the latter for proofs or a more detailed account of the concepts being used. Kunen's Set Theory draws a clear path towards forcing, which is the main topic of the book, so this way you will avoid reading things in the other book that could distract you from your main objetive. I hope this helps. Good luck!
If you have little experience with logic, I think it is good to read through all of part 2 of The Foundations of Mathematics, but if the purpose is purely in set theory / forcing, then you can easily skip all parts related to proof theory.
However, as the other answer states, Set Theory is more focused and it is almost completely self-contained, so you could also go ahead and read only the second book. It does also explain the necessary model theory from scratch.
In Set Theory as well, there are parts you can skip if the main interest is forcing. In particular, the sections on Martin's Axiom go a lot deeper than is necessary, and there are parts about combinatorics that could be skipped (e.g. anything mentioning $\Diamond$), and I remember a lot of things about Foundation that are not very relevant to forcing.
I'd like to recommend two other books as well, namely
- Set Theory by Thomas Jech
- Combinatorial Set Theory by Lorenz Halbeisen (which is available on his webpage)
The first is generally a good reference book for Set Theory and the first part goes into more detail about certain concepts than Kunen's book does. The chapters about Forcing are quite concise, but Jech has a focus on treating forcing with a background of Boolean-valued Models. To some this may be more intuitive, and in general reading about both may help you understand better what forcing really does.
The second book has a focus on cardinal characteristics and combinatorics (infinitary Ramsey theory), but its chapter on forcing is very gentle and has many details worked out. It may be more friendly than Kunen's exposition.
