Talk:Axiom of determinacy

message: I just dropped a passage on infinite logic and the axiom of determinacy.  The particular version of infinite logic in which this can be done I'm not sure of.  One exists obviously but it may be inconsistent.  This is just because the axiom of determinacy may be inconsistent.  Barnaby dawson 17:54, 24 Oct 2004 (UTC)
replies: text: Cool, thanks for your additions! -- Schnee 20:34, 24 Oct 2004 (UTC)
thread_title: Infinite logic

message: I've written (well, started) a new Determinacy article, intended to be the central reference point for the new :Category:Determinacy, that treats determinacy in a more general context than just the axiom of determinacy.  It is not intended to supplant this article, but to complement it.  However it's possible that there's some content in Axiom of determinacy that really belongs in Determinacy, now that it exists -- I'll think about this later.  Maybe. --Trovatore 05:15, 2 September 2005 (UTC)
replies: text: Actually, I don't think any content needs to be moved out of Axiom of determinacy.  The "types of games that are determined" content will probably be redundantly treated in Determinacy, but it's short and has expository value, so it's OK where it is.  Mainly there's content that I thought needed to be added to Axiom of determinacy, that now ought to go in Determinacy instead. --Trovatore 05:19, 2 September 2005 (UTC)
thread_title: New [[Determinacy]] article

message: A list of a few things that come to mind that still need to be added, and belong in this article, not in the little Axiom of determinacy subsection of the Determinacy article.  (That subsection isn't written yet, but is expected to say what AD is, that it is provably false from ZFC, that it's true in L(R) (assuming large cardinals), and then point to this article for more detailed information).
# Consequences of AD for the alephs (e.g. ℵ1 and ℵ2 are measurable, ℵn for finite n≥3 are all singular with cofinality ℵ2.
# Nonexistence of an uncountable wellordered sequence of distinct reals, or even distinct countable sets of reals.
# The coding lemma (should probably have its own article, but there should be a blurb here).
# The projective ordinals, such as δ13 (idem).
# Models of AD, beyond L(R).
--Trovatore 16:45, 3 September 2005 (UTC)
thread_title: Additions still needed to this article

message: Okay, I don't understand this article.  At all.  In fact, I don't think I understand a single sentence beyond the first.  Of course, I'm a layman, so I can't expect to understand it fully without some set theory courses (unless it's a bit more basic than I think), but I could at least have some idea of a) what it implies for easily-understandable concepts, b) what direct or indirect practical applications it may have, and c) how important it is in its field, probably in addition to at least some other stuff.  Surely this concept is no more advanced than, say, quantum chromodynamics (at least the latter has a much fancier name ;)), but the latter has an introduction that's mostly understandable to high-school graduates. —Simetrical (talk) 03:25, 26 December 2005 (UTC)
replies: text: You must have gone to an unusual high school, but yes, I agree that this article could use a better intro (and not just from an accessibility point of view). Take a look at Determinacy#Basic notions and see if that helps any, and if there's anything from there you think might be imported to this article. --Trovatore 21:42, 28 December 2005 (UTC)
replies: text: I also found the article very confusing until I had a look at that link. What I'm going to do is change the link on the word "games" in the third sentence (the statement of the axiom itself) to point to that link instead of to "game theory". After all, what that link does is explain what sort of game this axiom is talking about, and that was just the missink link I needed to understand it. 91.105.63.245 (talk) 19:57, 20 April 2008 (UTC)
replies: text: Actually I don't agree with the link to game theory. Game theory is almost exclusively about games of imperfect information -- games of perfect information (at least, with only two players) are considered kind of trivial from a game-theoretic perspective. They may not be trivial in practice, of course, but the parts of them that are difficult are not interesting to game theorists qua game theorists. I can believe that the link to a section of determinacy may not have been ideal, but I think it's better than a link to game theory. --Trovatore (talk) 21:31, 20 April 2008 (UTC)
replies: text: Whoops, now I see that we're in "violent agreement". Sorry about that -- I read the diffs backwards. --Trovatore (talk) 21:55, 20 April 2008 (UTC)
thread_title: Template:Technical

message: Is there something known about what weaker forms of AC are consistent with AD? I would really like to see wether DC or something even stronger is compatible with AD or not, or is it undecidable that it is undecidable to decide? XD Scineram 15:48, 1 September 2006 (UTC)
thread_title:  Choice 

message: The article says:
: It follows from the existence of sufficient large cardinals that ... and that AD holds in L(R).
How can this be when, in L(R), L(R)=L=V. Therefore in such models AC holds, so AD cannot also hold? --Hccrle (talk) 09:47, 26 April 2008 (UTC)
replies: text: You have a false premise -- it's not true that L(R) satisfies V=L. For example L(R) contains 0#, and is able to see that 0# is not an element of L (but that it is an element of L(R)). --Trovatore (talk) 21:23, 26 April 2008 (UTC)
thread_title: On Types of Games That Are Determined

message: The article currently states:
:the axiom of determinacy implies that all subsets of the real numbers are Lebesgue measurable
This begs the question: what about the Banach-Tarski paradox? what about the Vitali sets? I guess that the statement "incompatible with CH" takes care of these, but a more explicit development would be nice.  linas (talk) 16:30, 8 May 2010 (UTC)
replies: text: Because AD implies that all sets of reals are measurable, it immediately follows that it refutes Banach-Tarski and the existence of the Vitali set.  I suppose that would be worth mentioning.  (Note that just because AD is incompatible with AC does not in itself imply that AD refutes Banach-Tarski or the Vitali set.) --Trovatore (talk) 17:26, 8 May 2010 (UTC)
replies: text: Hmm, actually, on rethinking it, I suppose I don't see an immediate argument from the measurability of all subsets of R to the measurability of all subsets of R3, which is what you need to directly refute Banach-Tarski.  But AD certainly proves the measurability of all subsets of R3, and the argument is going to be essentially identical to the proof for subsets of R.  --Trovatore (talk) 18:37, 8 May 2010 (UTC)
thread_title:  Measurablility? 

message: By the way, can someone (meaning Trovatore almost surely) look at the infinite action spaces section of Extensive-form game (the game described here is of that kind). Herrlich says that omega-long games and games where the action sets have omega cardinality are "complementary", but I'm not sure what that entails for the determinatness of perfect games of finite length but with infinite action spaces of various cardinalities. Tijfo098 (talk) 06:23, 27 March 2011 (UTC)
replies: text: I have responded at that article's talk page. --Trovatore (talk) 09:50, 27 March 2011 (UTC)
thread_title:  Help requested at extensive-form games 

message: Hello fellow Wikipedians,
I have just modified  on Axiom of determinacy. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
*Corrected formatting/usage for http://www.ams.org/journals/jams/1989-02-01/S0894-0347-1989-0955605-X/S0894-0347-1989-0955605-X.pdf
When you have finished reviewing my changes, please set the checked parameter below to true or failed to let others know (documentation at ).
Cheers.—cyberbot IITalk to my owner:Online 04:41, 27 May 2016 (UTC)
thread_title:  External links modified 

message: I have noticed that the original versions (Mostowski45, Mycielski64, Jech77, Herrlich06) talk of an axiom of determinateness. The only proper literature I found so far on axiom of determinacy is Jech05, there the author actually uses both names from what I dug up without a real preference or explanation of one over the other. Wouldn't the naming axiom of determinateness be more historically accurate? In my opinion it also makes it easier to distinguish between determinacy as a property of finite games and determinateness as an axiom of infinite games. Pinky Suavo (talk) 18:34, 21 March 2017 (UTC)
replies: text: "Determinateness" is the older name, but in my experience that's not what people call it in more modern work. --Trovatore (talk) 19:50, 21 March 2017 (UTC)
thread_title:  Naming: Determinacy vs Determinateness 

message: Hello. I have added a second proof of incompatibility with AC. In my opinion, this addition adds to the article beyond the existing proof using the well-ordering of the continuum. (Speaking for myself only, I barely understand that proof.) This addition could make this article more understandable to non-experts. This has connections with other strategy-stealing arguments. My direct knowledge of this proof is from https://www.youtube.com/watch?v=Kj5RCs1FHcc (apologies if improper citation), a video by math PhD Youtube: SackVideo. (I otherwise do not know him.) He offered the citation "Proposition 28.1 of The Higher Infinite by Akihiro Kanamori," which I do not have access to. I quote Sack: "This argument actually shows something slightly stronger than AD and AC are incompatible: It shows that AD implies there are no non-principal ultrafilters over ω.  (In the video, we use AC to construct a non-principal ultrafilter. However, the existence of a non-principal ultrafilter is weaker than AC.)" This could maybe be added to this article, but I will not do so due to lack of familiarity.
2600:1700:2CC1:6C20:7439:48EF:47E3:7A0A (talk) 19:10, 23 August 2023 (UTC)
thread_title:  Second proof of incompatibility with AC 

message: In the "2d proof" mentioned in the comment immediately above, added on 23 Aug 2023, I see this text:
:Observe the equivalence relation on \{0, 1\}^\omega  such that two sequences are equivalent if and only if they differ in a finite number of terms. This partitions the set into equivalence classes, and let T be the set of equivalence classes (such that T has the cardinality of the continuum). Define \{0, 1\}^\omega \rightarrow T that takes a sequence to its equivalence class.
Does this equivalence relation have a name? I realized earlier today that this equivalence relation is more or less exactly the same thing as the Vitali set. The correspondence is "obvious": The Vitali set asks you to consider equivalence classes of reals that differ by a rational. The above asks you to consider equivalence classes of reals that differ by a dyadic rational. But the rationals and the dyadic rationals are in one-to-one correspondence, via the Minkowski question mark function. So these are constructing the same thing.
Just to be extra-super clear about what I am saying, above: the \{0, 1\}^\omega  is of course the Cantor space, and it can be mapped onto the reals in the usual way, as infinite binary string expansions. Infinite binary strings that differ in a finite number of places map to reals that differ by a dyadic rational (since dyadic rationals are in one-to-one correspondence with finite-length binary strings.) So the equivalence relation is just \{0, 1\}^\omega / \mathbb{D} with \mathbb{D} the dyadic rationals, while the Vitali set is just \{0, 1\}^\omega / \mathbb{Q} (and Vitali also asks you to pick one and only one representative from each equivalence class). And \mathbb{D} and \mathbb{Q} are in 1-1 correspondence via Minkowski question mark. There's a third article talking about this construction: the article on the infinite parity function describes the same thing (but badly) although it does mention there are uncountably many different infinite parity functions (one assignment per equivalence class).
So I am seeing this construction in three different articles in 24 hours, and I'm wondering "does it have a name"? In my mind, I'm going to call it a "Vitali-type construction" but this is my personal neologism. Can anyone do better? 67.198.37.16 (talk) 05:54, 28 November 2023 (UTC)
replies: text: Yes, that equivalence relation (usually on {}^\omega\omega rather than {}^\omega 2, but it doesn't make much difference) is known as E_0.  It's very special in that it's the smallest Borel equivalence relation (in the order of Borel reducibility) that does not admit a Borel function giving a real-valued complete invariant for the equivalence relation.  See this paper by Harrington, Kechris, and Louveau. --Trovatore (talk) 08:15, 28 November 2023 (UTC)
replies: text: Thank you! Perhaps it would be appropriate to start an article on E_0? Or more directly, I'd be interested in such an article, even if its a stub. 67.198.37.16 (talk) 21:05, 28 November 2023 (UTC)
replies: text: "... that does not admit a Borel function giving a real-valued complete invariant for the equivalence relation." I assume that should read "in the standard model of the reals" since presumably one can still consistently assign it a measure of epsilon in non-standard analysis. But perhaps all of my comments should be moved to StackExchange, instead of here. Noted. 67.198.37.16 (talk) 14:35, 29 November 2023 (UTC)
replies: text: This is an enjoyable discussion and I wish you'd ask a question on WP:RD/Math, where we could continue it.  But briefly, no, nonstandard analysis has no loopholes to offer here. --Trovatore (talk) 17:22, 29 November 2023 (UTC)
replies: text: I'll ask when I figure out what the question really is; I will need to do more reading. In the meanwhile, I will conclude that the relevant keywords are the Glimm-Effros dichotomy, the Feldman–Moore theorem and the Lusin–Novikov uniformization theorem, with at least two of those redlinks, if not all three, being redirects to Countable Borel relation which states them. 67.198.37.16 (talk) 19:13, 1 December 2023 (UTC)
replies: text: Thanks for pointing me to that last article.  The right division of labor for articles in this area is a little hard to work out.  I don't think I'd put so much of it into countable relations specifically (there's an article I started years ago on Borel equivalence relations, a little more than a stub, but I think it should probably host much of this content).  Anyway it sounds like I should get in touch with , except that he/she seems to have submitted that article for creation and then disappeared. --Trovatore (talk) 19:30, 1 December 2023 (UTC)

text: And while we're on the topic, the run-length encoding from Cantor space to Baire space, does it have a common name used in set theory? I call it "the Minkowski question mark function", because that is what it is after one projects Cantor/Baire down to the reals. But, before doing this projection, calling it the question mark is perhaps a mild neologism? 67.198.37.16 (talk) 22:29, 28 November 2023 (UTC)

text: Hi @Trovatore above you said "this is an enjoyable conversation"; sadly, I cannot continue it, my focus has gyrated to somewhere else. However, while closing out open browser tabs, I wanted to mention that I converted the above late-night ruminations into an 18-page PDF, here. It is an extended "homework exercise"; there is no ground-breaking content. I wanted to mention it as a closing remark. Unfortunately, I never had the chance to dig into Glimm–Effros dichotomy, etc. as I had hoped. Alas. Maybe later. 67.198.37.16 (talk) 18:31, 13 April 2024 (UTC)
thread_title:  Name for Vitali-type equivalence relation in second proof?