User blog comment:Edwin Shade/Can Chess Ordinals Produce Functions With Uncountable Growth Rates ?/@comment-30754445-20171223203423

The idea of these chess ordinals is really simple.

Say we're playing ordinary chess and you're playing with the white pieces. What does it mean if you have  a position that's "white to mate in 5"? I mean, apart from the fact that as your opponent I'm in big trouble?

It means that you can gurantee - regardless of my future moves - a win in 5 moves. More specificially, it means that you can gurantee that on your next turn, you'll get a position of "white to mate in 4" (at worst).

And even more specifically, "mate in 5" means this: You can gurantee that on your next turn, you'll get a position of "white to mate in exactly X" for some number X<5.

So, what does "white to mate in ω" means in infinite chess? It means exactly the same thing: You can gurantee that on your next turn, you'll get a position of "white to mate in X" for some X<ω. In other words: You can gurantee than on your next turn, you'll get a position of "white to mate in n moves" for some finite number n (which is simply a position where white can force a win in n moves).

Note that you cannot gurantee any specific value of n. When confronted with "white to mate in ω", I - as your opponent - can make n as large as I wish on my very next move. That's why it is "mate in ω" and not "mate in n+1" for some given n.

If you look at the specific example given in the PBS video, you'll see that black can - by moving his rook n spaces upwards - postpone his defeat for 2n moves. Since this is played out on an infinite board, I (as black) can move the rook as far as I wish and make n as large as I want. I could, for example, move the rook a billion spaces, which will leave you with a winning position of "white and mate in two billion moves" (as you can imagine, infinite chess has no time controls and it also assumes both players are immortal and very patient). But once the rook has moved, I'm doomed. Once n is set, there's no turning back. It becomes an ordinary situation of "mate in x moves", and you already know how these situations end.

So that's what ""white to mate in ω" means.

And for larger ordinals, the situation is exactly the same:

"White to mate in x" for some ordinal x, means that he can gurantee getting a position of "White to mate in y" for some ordinal y<x.

So if you are in a position of "White to mate in ω2 moves", this means that you can force me to hand you a position of "White to mate in ωxn+m moves" on your next turn. I - as black - could make n as large as I please, but once I've set the value of n then I'm stuck with that decision and the ordinals will keep decreasing turn after turn until

Now here's the really interesting thing:

Any position which has "White to mate in moves", if played to its conclusion, will result in white winning after a finite time. Even though we are dealing with transfinite ordinals, there is no way for black to postpone the checkmate forever.

This is due to a fundamental theorem about ordinals:

any decreasing sequence of ordinal must - eventually - terminate.

(amazingly enough, this is true even for a sequence that begins with an uncountable ordinal)