A finite portion of the chess position with the largest currently known ordinal value, \(\omega^4\).[1]
\(\omega_1^\mathfrak{Ch}\) (pronounced omega one of chess) is a large countable ordinal, defined like so:[2][3]
- Consider the game of chess played on a (countably) infinite board. Only a finite number of pieces are allowed.
- Consider the set of all positions in infinite chess \(P\) and define a function \(\text{Value}: P \mapsto \omega_1\) like so:
- If White has won in position \(p\), then \(\text{Value}(p) = 0\).
- If White is to move in position \(p\), and if all the legal moves White can make have a minimal value of \(\alpha\), then \(\text{Value}(p) = \alpha + 1\).
- If Black is to move in position \(p\), and if all the legal moves Black can make have a supremum of \(\alpha\), then \(\text{Value}(p) = \alpha\).
- \(\omega_1^\mathfrak{Ch}\) is the supremum of the values of all the positions from which White can force a win.
There are a few variants of this ordinal:
- If an infinite number of pieces are allowed, the supremum is called \(\omega_1^{\mathfrak{Ch}'}\).
- With 3D chess, the supremum is called \(\omega_1^{\mathfrak{Ch}_3}\).
- With 3D chess with an infinite number of pieces, the supremum is called \(\omega_1^{\mathfrak{Ch}_3'}\). This ordinal has been proven to equal the first uncountable ordinal.
Evans and Hamkins proved that \(\omega_1^\mathfrak{Ch}\) and \(\omega_1^{\mathfrak{Ch}_3}\) are at most the Church-Kleene ordinal, and \(\omega_1^{\mathfrak{Ch}'}=\omega_1\). Although it has not been proven, it is believed that some of these ordinals are as large as possible — that is, \(\omega_1^\mathfrak{Ch} = \omega_1^{\mathfrak{Ch}_3} = \omega_1^\text{CK}\). Note it has been proven that \(\omega_1^{\mathfrak{Ch}'} = \omega_1\).
Video[]
Here is a video which explains Infinite Chess in a less formal way that should give you an idea of how chess ordinals are calculated.
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Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function) · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function · Arai's psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · Berkeley cardinal · (Club Berkeley cardinal · limit club Berkeley cardinal)
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)