User blog comment:Edwin Shade/Can Chess Ordinals Produce Functions With Uncountable Growth Rates ?/@comment-1605058-20171222153040/@comment-1605058-20171222221034

This reasoning is not sound. Compare it with the following example, based on exactly the same idea: Consider the supremum of lengths of well-orderings which we can construct on the set with \(n\) elements (which is \(n\)), respectively on the set with \(\omega\) elements (which is \(\omega_1\)).

As I've said, with infinite number of elements, we can do strictly better than with any finite number of elements. The reason is that, intuitively, a well-ordering of \(\omega\) elements cannot be viewed as the "limit" of its finite pieces. Similarly, in chess, an infinite position cannot be always viewed as the "limit" of its finite parts. For a simple illustration of this (without invoking ordinal values) consider the infinite position with an infinite vertical line of white pawns, a couple of white queens on one side and a black king on the other side. Clearly this position is drawn, but if we consider any finite subset, queens can go around and mate the king.