Talk:Determinacy

message: I can easily imagine someone stumbling across this page and thinking it's an unnecessary duplication of Axiom of determinacy, an article of long standing, and ask why I don't simply try to improve that page.
Well, that was my thought originally, but I was running into all sorts of difficulties figuring out where things should go.  Axiom of determinacy didn't have a good definition of "game", "strategy", or "winning strategy", and it wasn't clear where to put them.  There's a winning strategy article, but it wants to be useful to game theorists as well, which was too much of a constraint.
Moreover, lots of times one wants to talk about determinacy but is not, in context, particularly interested in the axiom of determinacy.  In particular, several pages had links that looked like determinacy, when AD was perhaps not the main point of interest.
So it occurred to me that there should be a central article about determinacy in general.  This is it, or rather the start of it.  My vision is that this article will give the basic notions and say something about how they stand in relation to each other.  Then there will be links out to more detailed articles on particular topics, such as axiom of determinacy, axiom of real determinacy, projective determinacy, long games, Blackwell games, axiom of quasideterminacy.
This article should be the central reference point for the new :Category:Determinacy.  If a major aspect of determinacy theory has been left out, then ideally in addition to writing an article on that particular topic, a new section or subsection should be added to this article, describing the topic briefly and saying how it relates to the other determinacy topics.
--Trovatore 05:46, 2 September 2005 (UTC)
thread_title: Rationale for this page

message: This page needs some help to make it more accessible.  I have a semi-math background (CS + statistics) and I have no idea what is going on.  This needs a bit of a gentler intro (like the one for NP vs. P article which is very good ) novacatz 13:17, 13 January 2006 (UTC)
replies: text: Well, that's what I had in mind for the Determinacy#Basic notions section. That strikes me as being of roughly the same level of technicality as the lead section of Complexity classes P and NP. Would you look over the "Basic notions" section again and see if you agree, and if not, let me know where it is you get lost? (As for the rest of the article, I wouldn't really expect it to be accessible to anyone without a set theory background; it's an inherently technical subject.)  Thanks, Trovatore 00:13, 14 January 2006 (UTC)
replies: text: Currently, the basic notions section reads like a math textbook (cf. 'firstly we shall consider...' and 'More formally, ... , recall that the latter consists of...').  If you look at the gentle introduction of NP vs. P article intro, you will see they spend a lot of time explaining what is the problem using layman English (cf. the 'time space tradeoff' has an explanation what 'time' mean and what 'space' means.  You can see that the page has a lot of somewhat loose descriptions that are useful on helping people grasp why the area is interesting (eg. 'NP Complete problems can be loosely described as...').  For determinacy - the intro is a bit weak on why the field is important (some examples might help) and starts getting into the results without explaning why they are significant (eg. determinacy from ZFC).  Someone with some elementary set theory knowledge (me) can appeciate this result a lot more if some explaination is provided on interepetations of the result. novacatz 04:58, 14 January 2006 (UTC)
thread_title: Too technical

message: Aleph4 pointed out at Talk:Axiom of choice that determinacy for games on arbitary sets implies the axiom of choice, and also implies that a countable union of countable sets is countable. The second result can be modified to show in AD that a countable union of countable sets of reals is countable.  These facts are very interesting to me and ought to appear here.  Also, Martin's later paper on Borel Determinacy would be good to reference; it is much easier to read than his first one from the Annals.  I will get to this sometime, so you can view this message as a reminder to myself as much as a prod to everyone else. CMummert 12:09, 6 October 2006 (UTC)
thread_title:  Discussion from the AC page 

message: I want to suggest making an article on Gale-Stewart games, which would be an expansion of the Games section here.  Then this article could just have a summary of the topic with a link to the main article.  The topics for the Gale-Stewart games article would be:
* The definition of a Gale-Stewart game on X^\omega where X is any set (as in Martin's paper on Borel determinacy).  The specific cases for Cantor and Baire space.
* The definition of a winning strategy.
* A statement of the theorem that closed games are determined.
* A link to Determinacy.
I am willing to do the writing; I want to see if there are objections before I do it. CMummert 17:37, 15 November 2006 (UTC)
replies: text: Well, I guess my concern is, what exactly is a Gale–Stewart game? One with a clopen payoff set? Is this really standard terminology? Gale's and Stewart's result is key, of course, but I really don't recall coming across this nomenclature for the games. --Trovatore 19:48, 15 November 2006 (UTC)
replies: text: A Gale-Stewart game is just a two-player game of perfect information of length ω.  This is standard terminology in some parts of the world; a google search turned up quite a few papers such as    that use the term without further citation. From JStor, I found that Jockusch uses the term (The Journal of Symbolic Logic Vol. 38, No. 2 (Jun., 1973), pp. 293-294) and so does Becker (The Journal of Symbolic Logic  Vol. 50, No. 1 (Mar., 1985), pp. 110-122).  So I think it is standard. CMummert 20:35, 15 November 2006 (UTC)
replies: text: Hm, still not too excited about giving the concept a name I'd never really heard for it. This is the content I'd intended to treat in the "games played on trees" section of the article. --Trovatore 21:36, 15 November 2006 (UTC)
replies: text: The idea I had in mind is to separate the material on games (which don't require strong axioms to define and are of interest in ZFC) from the material on determinacy beyond ZFC, which is more esoteric. Since Gale-Stewart game is a standard term describing these games, that was the title I suggested, but any other title is OK with me. The present article, if all the sections were filled in, would be quite long and intimidating. CMummert 22:02, 15 November 2006 (UTC)
replies: text: The article might be a good idea. I think it would be unfortunate, though, if it presented ZFC as some sort of natural dividing line. I really don't think it is. --Trovatore 07:07, 9 December 2006 (UTC)
thread_title:  Split out [[Gale-Stewart game]]s? 

message: I think the angle brackets should be written as "\langle" and "\rangle" instead of "<" and ">". --Trigamma 18:15, 8 December 2006 (UTC)
replies: text: I don't like \langle and \rangle; never have.  They are too tall, and the angle is too obtuse.  It would be nice to have something a little bit in between. --Trovatore 18:46, 8 December 2006 (UTC)
replies: text: My problem is that < and > are typeset the wrong way: they are not treated as delimiters but as binary relations. You can see that in the first formula, where the closing > is next to the \in. But by the way: Why do you use angle brackets at all and not round parentheses? --Trigamma 19:54, 8 December 2006 (UTC)
replies: text: Um. In my experience, angle brackets are just more usual, in set theory, as a way to indicate tuples, than are round brackets. I don't really know why this should be so. Maybe they have fewer other meanings (for example, (αβ) could be the pair <α,β> or the open interval from α to β). --Trovatore 21:09, 8 December 2006 (UTC)
thread_title:  Angle brackets 

message: Quote from the article:
"Martin's proof uses the powerset axiom in an essential way. There is a level-by-level result detailing what fragment of the powerset axiom is necessary to guarantee determinacy through what level of the Borel hierarchy."
I am curious about what "fragment of the powerset axiom" means more precisely. Does it mean that in this and that model, built not using the full power set operation but something that is possibly weaker (like when L is built compared to when V is built), then determinacy is guaranteed through a lower degree inte the model? YohanN7 (talk) 14:47, 14 September 2009 (UTC)
replies: text: It means: how many iterations of the powerset axiom are required in order to guarantee the desired amount of determinacy. In other words, there is result giving the minimum necessary α for Vα to be a model of the desired amount of determinacy. Now the powerset axiom itself says nothing about iteration; it is really the axiom of replacement that is used to ensure that the ordinals in the model are long enough. — Carl (CBM · talk) 15:18, 14 September 2009 (UTC)
replies: text: Ah, that explains it. Would you mind if I attempt a rewording of that passage in the article? YohanN7 (talk) 16:37, 14 September 2009 (UTC)

text: It appears to be done already. Things happen quickly here;) YohanN7 (talk) 16:49, 14 September 2009 (UTC)
thread_title:  Determinacy from ZFC 

message: Yes it does. At least in the sense that if both players actually have strategies as described in the article, then the recursion theorem guarantees that there is a sequence f:ω->ω whose values can be matched against the lookup table. This escaped me when I first read the article (quite some time ago). Perhaps the article could point out that (and why) the infinite games end?
YohanN7 (talk) 19:57, 5 April 2011 (UTC)
thread_title:  Does an infinite game ever end? 

message: Some of my edits adding inline LaTeX were reverted because of "sizing issues". However, there are still sections where inline LaTeX is used despite the objections to my edits with respect to "sizing issues".
While the issue is mainly aesthetic, I think that typesetting consistency would aid the readability of the article.
2601:8:AB80:7C0:573:A3AB:E740:805B (talk) 01:55, 20 May 2014 (UTC)
replies: text: Yeah, I noticed that later.  Those should be converted to HTML as well.  I'll try to find some time and take care of it. --Trovatore (talk) 02:03, 20 May 2014 (UTC)
replies: text: Done 2601:8:AB80:7C0:573:A3AB:E740:805B (talk) 02:09, 20 May 2014 (UTC)
replies: text: Thanks! --Trovatore (talk) 02:42, 20 May 2014 (UTC)
thread_title:  Inline LaTeX 

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thread_title:  External links modified 

message: Can one game within a class be determined, and another not? If so, in "Determined games" we should revise:
:A (class of) game(s) is determined if for all instances of the game there is a winning strategy for one player (not necessarily the same player for each instance), or a drawing strategy for both players.
to:
:A game is determined if there is a winning strategy for one player, or a drawing strategy for both players.
The latter is simpler and correct. The former may be too broad and I think not always strictly true.LithiumFlash (talk) 01:44, 27 March 2017 (UTC)
replies: text: "Class" here means pointclass.  Specifically one closed downward under Wadge reducibility, and maybe with some other properties that aren't specified too carefully. --Trovatore (talk) 02:03, 27 March 2017 (UTC)
replies: text: Understood. If that is the case, after clarifying and recasting the sentence, the article should say so. But I won't change anything yet at this point. Let's see if there is other commentary. Then we can decide if we should make this point in a way that is simple, general, and to the point. Or be more specific, carefully amending the sentence with the restricted definition as you mention. Thanks.LithiumFlash (talk) 02:18, 27 March 2017 (UTC)
thread_title:  What is definition of a "class" of games? 

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thread_title:  External links modified