Has this ordinal been discovered by you, FB100Z? Ikosarakt1 (talk ^ contribs) 22:06, October 1, 2013 (UTC)
- Also, it would be pretty interesting what would happen if we extend chess up to 4 and more dimensions and allow infinite number of cells. I guess that n-dimensional chess with infinite number of cells has limit ordinal \(\omega_{n-2}\). Ikosarakt1 (talk ^ contribs) 22:12, October 1, 2013 (UTC)
- The \(\omega_1\) of chess arose from a question by Johan Wastlund on MathOverflow, and addressed in a paper by C.D.A. Evans and Joel David Hamkins, who coined the term \(\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}\). The paper is called “Transfinite game values in infinite chess.” and is available here: http://arxiv.org/abs/1302.4377
- Note that, even in n-dimensional chess with an infinite number of cells, there are countably many moves that can be made by each player from any position. So the ordinal of that postion is a countable supremum of previous ordinals, so all positions are associated with a countable ordinal. Since \(\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}{}_3}\) = \(\omega_1\), it follows that \(\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}{}_n}\) = \(\omega_1\) for all n.
- FB100Z, I'm not seeing any difference between \(\omega_1^\mathfrak{Ch}\) and \(\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}\). How are they different? Deedlit11 (talk) 23:00, October 1, 2013 (UTC)
- Different in notation, or in definition? One should have a tilde under the Ch and the other does not. 01:59, October 2, 2013 (UTC)
- FB100Z, I'm not seeing any difference between \(\omega_1^\mathfrak{Ch}\) and \(\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}\). How are they different? Deedlit11 (talk) 23:00, October 1, 2013 (UTC)
- IIRC, this little tilda means that we allow infinitely many pieces to appear on a board. LittlePeng9 (talk) 05:30, October 2, 2013 (UTC)
- This ordinal is not my invention; I got it from Cantor's attic. FB100Z • talk • contribs 01:59, October 2, 2013 (UTC)
Fundamental sequence? you're.so.pretty! 18:58, May 30, 2014 (UTC)
- For finite setups, it's simple - for n-th element of fundamental sequence we look at all patterns fitting in nxn square. For others - I have no clue. LittlePeng9 (talk) 19:29, May 30, 2014 (UTC)
Simpler games[]
One problem with the mathematics of chess is that it's mathematically rather arbitrary. Since I had a weird obsession with variant chess years back, I propose a modification. Define an m,n-leaper as a chess piece that can make a jump by adding the vector <±m,±n> or <±n,±m> to its current position, possibly capturing an enemy piece at its destination. A knight is therefore a 1,2-leaper, and a king is a combination of a 1,0-leaper and a 1,1-leaper (ignoring royalty conditions). Define an m,n-rider as a chess piece that, for all k > 0 and all v of the form <±m,±n> or <±n,±m>, can move or capture kv provided that jv is unoccupied for all 0 < j < k. Thus a rook is a 1,0-rider and a bishop is a 1,1-rider, and a queen is a combination 1,0-rider and 1,1-rider. This generalizes in an obvious way to higher dimensions. it's vel time 16:27, September 25, 2014 (UTC)
Better bounds[]
The current lower bound is \(\omega^4\). Can we do it to beat it? AarexWikia04 - 20:21, August 10, 2016 (UTC)
- Feel free to try. LittlePeng9 (talk) 05:45, August 11, 2016 (UTC)
- There's one already: \(\eta_0\). 80.98.179.160 16:24, November 14, 2017 (UTC)
- He is either bluffing or has misunderstood Aarex's question, as, with all due respect, 80.98.179.160 is known by now for making unsupported claims.
-- Edwin Shade 21:07, February 7, 2018
Clarifying Ordinal Representations[]
The fact that the ordinals \(\omega_1^{ {\mathfrak{Ch}_{\!\!\!\!\sim} }_3}\), and \(\omega_1^{\mathfrak{Ch}_3}\), (the supremum of third-dimensional chess with an infinite number of pieces and the supremum of third-dimensional chess with a finite number of pieces respectively), are nearly identical aesthetically has righfully caused confusion among people - even to those who are knowledgeable about infinite chess !
Therefore I propose the notation be changed, and apostrophe's be adopted, so therefore \(\omega_1^{ {\mathfrak{Ch}_{\!\!\!\!\sim} }_3}\) will be \(\omega_1^{\mathfrak{Ch}_3'}\), (pronounced "omega one of chess-prime"), a better option. I am posting this message in the talk page because I realize that changing the notation of a professional research paper might be seen as controversial, but I really feel it is for the better of anyone who wants to understand the matter better.
If you have any objections please leave them in response to this comment, in the meanwhile I will be changing the article to conform to this less confusing convention.
-- Edwin Shade 21:03, February 7, 2018
Location on List of Countable Ordinals[]
If this ordinal isn't proven to be larger than \(\omega^4\), should it be moved down on the page? It's possibly as large as \(\omega_1^{CK}\), but without tighter lower boundsC7X (talk) 15:22, 25 January 2021 (UTC)