Timeline for What surreal numbers are representable by Red-Blue Hackenbush games?
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Aug 7, 2018 at 13:29 | comment | added | Paolo Lipparini | Probably thie above works even in the case when we keep something above $x$, provided it is connected to the ground in another way., but I have not completely looked at the details. I think that in any case if a game does not eventually ends, we end out wiht an infinite antichain anyway. | |
| Aug 7, 2018 at 13:27 | comment | added | Paolo Lipparini | In general I think we need well partially ordered set (wpo's) to define Hackenbush in full generality, since a play on an infinite antichain has no end. A wpo is a well founded ordered sets without infinite antichains. There is a characterization saying that a wpo is a partially ordered set such that there is no infinite chain $x_1$ $x_2 \dots$ such that for $i < j$ either $x_i < x_j$ or $x_i, x_j$ are incomparable. This means exactly that any game eventually ends, provided a move means picking some $x$ and discarding everything which is above $x$. | |
| Aug 29, 2017 at 18:06 | comment | added | Joel David Hamkins | Your example of side-by-side chains with another on top are an example of a well-founded relation with colored adjacencies. But I am not clear on whether one wants that when you cut one of the lower chains, do the nodes above it on that lower chain stay connected to the ground via the upper chain? Or did you intend that the tail of the lower chain fall when one its edges is cut, even if the upper chain remains? | |
| Aug 29, 2017 at 16:17 | comment | added | Will Sawin | @TimothyChow It's a bit strange to ask and answer a mathematical question about a mathematical object that is not completely well-defined, but of course you're completely right that Conway has pinned down the definition well enough to answer this question. | |
| Aug 29, 2017 at 15:36 | comment | added | Timothy Chow | That might be too much of a stretch, though. Anyway, I believe that this is uncharted territory and we're free to make up our own definitions. | |
| Aug 29, 2017 at 15:35 | comment | added | Timothy Chow | @WillSawin : As I said, I'm not sure anyone has defined Hackenbush in "full generality." If we were to try, I think that at minimum, we'd want (a) the moves to be represented by red and blue edges in a graph, (b) games to terminate after finitely many moves, and (c) some notion of being "connected to the ground". The last of these admits some room for debate over the precise meaning, e.g., one could imagine three copies of $\omega$, two of them side by side on the ground, and the third copy perched atop both of them, to be disconnected only if both lower copies of $\omega$ lose an edge. | |
| Aug 29, 2017 at 15:23 | comment | added | Joel David Hamkins | What is the right formal definition of a red-blue Hackenbush game? Having ordinals is fine, but it would seem one would also want well-founded trees, whose adjancency relations are colored. But there is no need to insist on a tree order. Well-founded relations with colored adjacencies? Or what? | |
| Aug 29, 2017 at 13:24 | vote | accept | swensonj | ||
| Aug 29, 2017 at 6:09 | comment | added | Will Sawin | @TimothyChow You're right, I didn't realize that - I either read about this a long time ago and forgot about it, or read about Hackenbush not in ONAG but in other works of Conway where this point wasn't stressed. I edited my answer so that it is now a correct answer to a different question. Still, one point confuses me - what is the formal definition of Hackenbush in general? Is it just a finite graph with finitely many ordinal branches added? | |
| Aug 29, 2017 at 0:59 | comment | added | Timothy Chow | I'm not sure that Conway ever gave a formal definition of Red-Blue Hackenbush in full generality. Hackenbush is supposed to be a lighthearted recreation rather than a cornerstone of the theory. However, it's clear from ONAG and The Book of Numbers that, at minimum, arbitrary ordinals are fair game. | |
| Aug 29, 2017 at 0:20 | comment | added | Timothy Chow | @WillSawin : Ah, I think I see; by a "rooted graph" you mean something that has no structure beyond that of a rooted graph right? In that case I agree. But I don't believe that swensonj's brief description of Red-Blue Hackenbush was supposed to be a formal definition of the game for someone who has no prior knowledge of it; I took it as an informal reminder. That is, I interpret "is a" in "a Red-Blue Hackenbush game is a rooted graph" in the same way that I interpret "is a" in "a Banach space is a vector space". It has that structure along with possibly further structure. | |
| Aug 28, 2017 at 21:19 | comment | added | Gerald Edgar | Hackenbush positions can have branches described by ordinals. So in that sense they are not "graphs". | |
| Aug 28, 2017 at 19:24 | comment | added | swensonj | I used the expression "rooted graph" in my question to mean a graph in which one vertex has been declared to be the "root." Of course this represents the ground: when a player deletes a cut edge, the game reduces to the connected component of the root. | |
| Aug 28, 2017 at 18:37 | comment | added | Will Sawin | The order is not determined by the edge structure as soon as your ordinal is greater than $\omega$. So you would be playing, instead of a game on graphs, a game on some generalized objects. | |
| Aug 28, 2017 at 18:14 | comment | added | Timothy Chow | @WillSawin : I'm not sure what you mean by a "rooted graph" but what I described is a valid Red-Blue Hackenbush game. It is a graph in the sense that it consists of vertices and edges, and it is rooted in that there is no ambiguity which vertex is the smallest in the total order that we impose. | |
| Aug 28, 2017 at 6:04 | comment | added | Will Sawin | But this isn't really a rooted graph, right? | |
| Aug 28, 2017 at 3:01 | history | answered | Timothy Chow | CC BY-SA 3.0 |