9
$\begingroup$

I just finished a first course in Descriptive Set Theory using Kechris' "Classical Descriptive Set Theory" and was hoping to find a good source for learning some of the Effective DST. Kechris doesn't mention it at all, and after looking at Moschovakis' "Descriptive Set Theory", it doesn't seem like the easiest thing to read. If that is the source I should use and I just need to suck it up and work through it, that is fine. But I would appreciate any advice on other places to look, even as supplements to a tougher standard source.

If it helps with the suggestions, I don't have any particular application or direction in mind. I just liked my course and wanted to continue learning in the area. Thanks.

Improve this question
$\endgroup$
2
  • 6
    $\begingroup$ Another option is Recursive Aspects of Descriptive Set Theory by Mansfield and Galen Weitkamp. $\endgroup$
    – 喻 良
    Commented May 14, 2012 at 3:41
  • $\begingroup$ I second Yu's recommendation, which is very clear and accessible. Sacks' Higher Recursion Theory is also a good source for effective DST. $\endgroup$
    – Ed Dean
    Commented May 14, 2012 at 5:49

2 Answers 2

Reset to default
13
$\begingroup$

A nice source is "Recursive aspects of descriptive set theory" by Mansfield and Galen Weitkamp, as mentioned by Yu in a comment. A problem with it is that it leaves out all the details of admissibility and its relatives. I also find it a bit more elementary than one would want, but it is a good initial source.

An excellent, encyclopedic reference is a handwritten manuscript by Alain Louveau. I do not know if Alain plans to turn this into a book, but it is very nice and the only source I know for some results. I imagine he may send you a copy if you email him directly.

Alekos Kechris and myself have been working on a book, covering what would be roughly a first semester graduate course on the subject. I am sorry to say that it has taken longer than it should, and it is entirely my fault, as suddenly there were too many things needing my attention at once, and I had to set this apart for a while. But there is a decent working draft, and I expect we should be completing it soonish.

(Once one is past this material, Moschovakis's book is certainly the place to go if one is interested in determinacy and related topics.)

Something that quickly becomes apparent is that a good working understanding of recursion theory is very helpful. Higher recursion theory may help as well, especially if one is interested in more advanced topics (such as applications to the theory of determinacy). Sacks's classical book is a very good introduction. You may also find useful the really nice draft by Chong and Yu, "Recursion Theory: From a generalized point of view", which you can find in Yu's page. (Let me reiterate that it is important to know higher recursion theory if one has interest in determinacy. Several results in the Cabal volumes were established in the language of higher recursion theory, and this shouldn't be an obstacle to understanding them.)

If you want to get started right away, I suggest you read the paper by Martin and Kechris, "Infinite games and effective descriptive set theory", Part 4 of "Analytic sets" by C.A. Rogers et al., Academic Press 1980. Developed from lectures given at the London Mathematical Society Instructional Conference on Analytic Sets, University College London, July 1978. 403-470.

Improve this answer
$\endgroup$
2
  • $\begingroup$ Suppose what I primarily want to know about is admissibility. What's a good source? $\endgroup$
    – Peter Gerdes
    Commented Apr 28, 2023 at 19:18
  • $\begingroup$ Bairwise's book seems to be the standard reference. He works with urelements, so some of his arguments can be simplified/shortened. Other good accounts are in books on higher recursion theory (Sacks, and the more recent by Chong and Yu, which also emphasizes the connections with descriptive set theory and forcing). Jensen has a paper on forcing over admissible sets, and S. Friedman also has several papers on this topic. $\endgroup$
    – Andrés E. Caicedo
    Commented Apr 28, 2023 at 21:17
2
$\begingroup$

I second Andres Caicedo's answer (and I'd really like to get my hands on Alain Louveau's notes! So if anyone possesses those...).

What I've found really helpful in conjunction with Kechris and Moschovakis are the lectures notes by David Marker (you can find them online). In contrast to Moschovakis they are not overly detailed and provide a nice overview, even of some "modern" results. But beware, you'll stumble upon quite some typing errors.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .