User blog comment:Alemagno12/Ordinals in CGT/@comment-5529393-20171007161637/@comment-25601061-20171007181950

Hmm.

Let's assume that {0,*|0,*} = *. {0,*|0,*}-* = {0,*|0,*}+* = {{0,*|0,*}+0,0+*,*+*|{0,*|0,*}+0,0+*,*+*} = {{0,*|0,*},*,0|{0,*|0,*},*,0} = {*,*,0|*,*,0} = {0|0} = *.

But since {0,*|0,*} = *, {0,*|0,*}-* = *-* = *+* = 0. This is a contradiction, so {0,*|0,*} can't be *.

And {0,*|0,*} can't be 0 either, since the first player can win, if both players play optimally.

So you are right