It is curious that Cantor Normal Form for ordinal numbers seems to arbitrarily stop at exponentiation in the hyperoperation hierarchy and not include tetration or beyond. The choice causes epsilon-nought to be the first ordinal not expressible with exponentiation. However, if tetration were permitted, then

epsilon nought = (tetrate omega omega)

Does including tetration induce weird holes or duplications, perhaps around (tetrate (omega+1) omega) or (tetrate omega (omega+1))? This could be similar to the difficulty of defining tetration for real second arguments.

Epsilon nought is supposedly useful for proofs using induction, so there is something deeper going on in the arbitrary choice to stop at exponentiation.

After including all operations in the hyperoperation hierarchy, one could define the first ordinal not accessible by an expression consisting of those operations. More precisely, one could define an ordinal that is the limit of (hyper1 omega omega) (hyper2 omega omega) (hyper3 omega omega) (hyper4 omega omega) (hyper5 omega omega) ..., so the omega-th member of the hyperoperation hierarchy.

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