User blog comment:Alemagno12/Analysis of Edwin Shade's ''three-symbol'' notation for ordinals/@comment-32876686-20171107013941

In answer to your question, [(][(][)]|[(][)][)] would be equivalent to <math\omega^{\omega^\omega}, but I'm not sure how you could expand it without knowledge of prior strings in my notation.

My notation was just meant to be an intuitive way to think of small ordinals, but as I have not yet formally defined it there may be some strings which can be resolved multiple ways without contradiction. (So at this point it is pretty arbitrary.)

For instance, should $$(||)=\omega$$ or $$(||)=\omega 2$$ ? As of yet, I have not explicitly defined how to handle these sorts of situations, but have relied on intuition for them. I am going to try a formal definition, but it may require more than three symbols.