Talk:Infinity-Borel set

message: Um, so actually I wrote this article, and no one else has touched it, so really I'm arguing with myself here.  But I thought about it and I'm not sure my definition is right.  On the other hand, if it is right, it's (I think) easier to follow than any alternative I can think of, so I don't want to jump the gun.  But I also don't want to mislead anyone.
Here's the issue; maybe someone can help me out:  The ∞-Borel sets are the ones that have ∞-Borel codes.  An ∞-Borel code is something that codes up the way you build an ∞-Borel set by starting with open sets (each basic open set gets a code), and then iterating the operations of complementation and wellordered union (there's a code for the complement of a coded set, and a code for the union of a sequence of coded sets).
Problem is--is the class of thus-coded sets in fact closed under wellordered union?  To prove it, you'd seem to need ACα, if your sequence of sets to be unioned up has length α, in order to choose a Borel code for each of the sets in the sequence, so you can form the new Borel code.
Any help appreciated. --Trovatore 04:21, 15 July 2005 (UTC)
thread_title: (Formerly) Disputed

message: So the definition now in place is harder to follow, and it's got a lot of LaTeX that simply cannot be displayed if anyone is expected to read the thing.  But it does have the advantage of being correct. --Trovatore 02:20, 16 July 2005 (UTC)
thread_title: Fixed

message: So I thought of another approach:  Dispense with the codes, and perform the transfinite iteration directly on the sets.  Something like this:
:For each ordinal α define by transfinite recursion Bα as follows:
:# B0 is the collection of all open subsets of X.
:# For a given even ordinal α, Bα+1 is the union of Bα with the set of all complements of sets in Bα.
:# For a given even ordinal α, Bα+2 is the set of all wellordered unions of sets in Bα+1.
:# For a given limit ordinal λ, Bλ is the union of all Bα for α<λ
:It follows from the Burali-Forti paradox that there must be some ordinal α such that Bβ equals Bα for every β>α.  For this value of α, Bα is the collection of ∞-Borel sets.
Whaddaya think?  Is it clearer than the definition with codes?  I did leave more out in this alternative version (e.g. I didn't really say what a wellordered union is, and it takes a little argument to see that clause 3. above gives you a set at all, rather than a proper class).  Also the code notion is important. --Trovatore 02:49, 16 July 2005 (UTC)
thread_title: Alternative definition

message: Would it be okay to make 'borel code' redirect to this? There is no article for it, and this one with with a few changes should suffice to explain the notion. Chimpionspeak (talk) 14:48, 20 February 2011 (UTC)
replies: text: Hmm, as a stopgap I suppose that would be OK, but it's not really an ideal solution.  This article is too specialized to serve the burden of explaining Borel codes in general.  It might be better to put a section on it into Borel hierarchy (or even Borel set) and then target the redirect to that section. --Trovatore (talk) 00:44, 22 February 2011 (UTC)
replies: text: Oh okay. Thanks. Chimpionspeak (talk) 07:55, 26 February 2011 (UTC)
replies: text: I have done some preliminary writing of it  here. I am a little hesitant about what more to add/say for fear of getting it wrong though! If you think this is okay, maybe we can shift it. Also, if you think something needs to be added, feel free.Chimpionspeak (talk) 16:47, 14 March 2011 (UTC)
replies: text: Honestly what we should do is use something from a published source.  It was a mistake on my part to roll my own in this article; that's not supposed to be how we do things here.  (I'm not going to actively remove it until I actually get around to finding one in a source, but if someone else did I couldn't really complain.) --Trovatore (talk) 18:27, 14 March 2011 (UTC)
thread_title:  Borel Code