User blog comment:Edwin Shade/Can Chess Ordinals Produce Functions With Uncountable Growth Rates ?/@comment-1605058-20171222153040/@comment-30754445-20171224063452

There are actually two different ordinals here which are notated by almost-identical notations:

  ω  C h  1       is the limit of the values of all positions with a finite number of game pieces.

  ω   C <span class="mi" id="MathJax-Span-153" style="font-size: 70.7%; font-family: MathJax_Fraktur;">h  <span class="mo" id="MathJax-Span-160" style="font-size: 50%; font-family: MathJax_Main;">∼    <span style="position: absolute; clip: rect(3.339em, 1000.43em, 4.167em, -1000em); top: -3.682em; left: 0.622em;"><span class="mn" id="MathJax-Span-161" style="font-size: 70.7%; font-family: MathJax_Main;">1       is the limit of the values of all positions, including positions with an infinite number of pieces.

The first one, obviously, must be strictly smaller than \(\omega_1\).