Talk:Wadge hierarchy

update reference
anyone mind if i update the "A Game for the Borel Functions" link in the references section? the link doesn't work in dare.uva.nl it's my own work so i'm not sure whether this is a breach of etiquette (changing signature) Drderevo (talk) 06:48, 11 November 2013 (UTC)

organization
So there are several related topics that could plausibly be article titles: plus the existing one, Wadge degree.
 * Wadge's lemma
 * Wadge reducibility
 * Wadge hierarchy
 * Wadge determinacy

I think they make perhaps two or three articles, with one "central" or "primary" one that introduces the subject and links to the "secondary" one(s) for more details. What I don't believe is that Wadge degree is the natural choice for the "central" article; I'd think that would be Wadge hierarchy, to show the big picture.

Just a thought for anyone (including me) who might be adding to this article—keep the big picture in mind, even if some of it is currently missing (that is, redlinks). Don't try to force everything under the title Wadge degree just because that's the article that currently exists. --Trovatore 16:33, 1 April 2006 (UTC)

went ahead and moved it
On second thought, I have my doubts that there's the necessary editor commitment to make more than one article anytime soon (certainly, there's not from me). All topics should probably be at least touched on at Wadge hierarchy, and we can make redirects-with-possibilities for them, to be expanded into articles if anyone gets ambitious. If Wadge degree itself were to be turned into an article, it should probably be about something like the fine analysis of Wadge degrees done by Louveau. --Trovatore 16:49, 1 April 2006 (UTC)

hierarchy(ies)
I've reverted the change from "a hierarchy" to "multiple hierarchies" (by an anon, unfortunately, so it's hard to find the editor to discuss it). Up to as far as determinacy holds, the Wadge order is linear, with the exception that non-self-dual Wadge degrees are incomparable with their duals. Also the standard term is "Wadge hierarchy", not "Wadge hierarchies".

To tell the truth I'm not even sure what "multiple hierarchies" is supposed to mean. Perhaps the editor who made the change would like to explain on this talk page. --Trovatore 01:39, 14 April 2006 (UTC)


 * Isn't there one hierarchy for each Polish space? Ben Standeven 17:13, 1 April 2007 (UTC)

True, but the Wadge hierarchy is viewed as a single abstract object that can be interpreted for each Polish space. Similarly, we often make reference to the family of Borel sets, even though there is a separate family for each space.Daniel Walker 15:28, 19 July 2007 (UTC)


 * Actually now that I think about it, there is possibly an issue here, at least if you look at it in a zoom lens. The Lipschitz degrees, at least, are different between Cantor space and Baire space -- in one of them there is exactly one self-dual degree between any two adjacent pairs of non-self-dual degrees, whereas in the other there are &omega; such self-dual degrees. Or something like that. I don't know whether there's such a distinction for the Wadge degrees.
 * And if you apply the continuous-function version of the definition of Wadge degrees to a connected Polish space, like the reals, things come out very different, at least at the low levels. When I was at North Texas this was a question some of us kicked around for a little while, but we didn't come up with anything terribly interesting.
 * Still, I think "Wadge hierarchy" is the standard term. --Trovatore 19:18, 19 July 2007 (UTC)

Proposed merger: fold Wadge game into this article
Wadge game is just a stub that is better as section in this article. I propose a merge with a redirect. Alternatively, fill out the stub and make it a real article. d a v i d w r 09f9(talk) 15:25, 19 May 2007 (UTC)

I agree.Daniel Walker 15:29, 19 July 2007 (UTC)

cleanup
I made some improvements (I think) to the page. My knowledge of the Wadge hierarchy is a bit limited and possibly incorrect, so someone with more experience should take a look.Daniel Walker 15:33, 19 July 2007 (UTC)

small changes
There were some mistakes in the page about the Wadge hierarchy that I corrected. About the discussion above, I think that the pages 'Wadge reducibility' and 'Wadge lemma' should be very short and just give the definitions, while there is more to say about the Wadge hierarchy. Also about Wadge determinacy there is quite some work and some open probems that I might add. Miaoku, 07-27-07

"Wadge theory"?
Would it make sense to move this article to Wadge theory? -- Karada 10:59, 27 July 2007 (UTC)


 * I completely agree that Wadge theory would be a better title. Miaoku, 27 July 2007
 * I don't. As far as I know it's not a term that's used. We try not to make up terminology. --Trovatore 16:15, 28 July 2007 (UTC)
 * Sorry but this is not made up. i think it was coined by Andretta. miaoku, 11 Aug 07
 * OK, I just don't recall hearing it. If Alex uses it in his actual papers (as opposed to just in conversation) I'm not going to object to the term. Still, that doesn't necessarily make it the best title. The question is whether the title should denote a field of study, or a mathematical object. Usually I lean to the latter. However, if there's enough content for say three or four articles, then it would make sense to have a central "spine" article called Wadge theory, that points to various subarticles, say Wadge degree, Wadge hierarchy, Wadge determinacy, Lipschitz degree, each of which is named for an object rather than a field of study. --Trovatore 22:54, 10 August 2007 (UTC)
 * I have also never heard the term Wadge theory, but a Google search shows a couple published papers that use the term. The difficulty with using new terminology like this is that it is difficult to say exactly what Wadge theory is. It's not clear to me that the current lede to this article is correct - wouldn't Wadge theory be the study of Wadge degrees, the Wadge hierarchy, and related notions such as Lipschitz degrees? The way the current lede is phrased, it seems to claim that e.g. Turing degrees are part of Wadge theory, which isn't right.  &mdash; Carl (CBM · talk) 23:09, 10 August 2007 (UTC)
 * I agree that the term Wadge theory is vague, I think it is used for studying some other types of degrees. Actually I haven't seen it in papers too, but I know a a guy who wrote his PhD thesis (with Andretta) about things related to the Wadge hierarchy uses this terminology..... your idea sounds good. I'm not sure at all what's a good title, I guess it depends on what else is going to be written... Miaoku 02:07, 11 August 2007 (UTC)
 * I'm not sure if there's enough material for more subarticles, so I suggest we keep the title as it is. Miaoku 10:24, 12 August 2007 (UTC)

removing tags
I'm thinking that at this stage it would be appropriate to remove the "in need of expert" and "clean-up" tags at the beginning of the page. What does everyone think?

Also, I'm putting in a "cite sources" tag. If someone with a better math library than mine could find specific sources for the claims made it would go a long way to "spiffying" this page up.Daniel Walker 17:43, 9 August 2007 (UTC)
 * I think the "cite sources" is not really appropriate -- it's not true that the article doesn't cite any sources. It cites a lot of them, in fact, though only one in the References section. Anyone who wants to go to the trouble of tracking down things for inline citation is more than welcome to do so, but I don't think ugly templates should be used as a stick to force that sort of thing. As long as all the claims we make are both true and available in the listed sources, that's the main thing.
 * On the other hand the article really does need a lot of cleanup, and I wouldn't be for removing that tag. When an article doesn't read cleanly and follow basic WP style, the reader should be warned that the article is being worked on.
 * The most obvious flaw is that the article currently has no lead section at all. After that, all the content is all in one section, in no evident particular order -- more or less a listing of facts without much context.
 * The selection of content is quite good, I think, and thank you very much for adding it! The organization, however, still has some distance to go. --Trovatore 18:08, 9 August 2007 (UTC)

changes
I just reorganized most of what I wrote earlier and added a little more. I apologize for any mistakes. I'm also removing the tags since I think (1) there are enough references, and (2) I tried to format everything. Since I'm new at wiki I'm not sure if this is good style so feel free to add tags again or make changes. I also thought from "references" you should be able to find most of the statements, and further reading is the rest, that's why I regrouped it. Does somebody want to write a page about "Rabin degree" etc? I don't know what it is... If not, would it be better to delete the links?

What about "Wadge degree" for a title? It seems that the word "hierarchy" suggests it's well founded (which is only the case in Baire space (and Cantor space etc.) and if you assume enough determinacy), while "degree" sounds more general to me... --Miaoku 02:22, 11 August 2007 (UTC)

In "See also" I have deleted "Veblen hierarchy" - although (a variant of) the Veblen function appears in computing the length of initial segments of the Wadge hierarchy, the Veblen hierarchy itself doesn't seem to be connected with the Wadge hierarchy. I also deleted "Rabin index", because there don't seem to be a lot of connections between Wadge degrees and this (although I don't know anything about the Rabin index, I just looked up its definition in a paper), please feel free to revert this if you think different. "Steel hierarchy" I also removed, since this seems to be a very specialized topic and I'm not sure if anything will be written(?) Miaoku 10:52, 12 August 2007 (UTC)

Removed the category: game theory, since in my view, there is (nearly) no connection between the games used in set theory and game theory in the usual sense. Miaoku 09:21, 22 August 2007 (UTC)