Abstract
It seems to be very difficult to construct a “natural” system of ordinal notations which would play the role for the theory of species that e0 plays for number theory and Takeuti’s ordinal diagrams of finite order play for the theory of iterated inductive definitions. However if one has the more modest aim of: (1) generating certain syntactical objects to be called ordinal notations, (2) defining a recursively enumerable predecessor relation between the ordinal notations, (3) proving in the theory of species that each ordinal notation is accessible with respect to the predecessor relation, and (4) showing the unprovability in the theory of species of the universal statement that all ordinal notations are accessible, then quite a simple solution can be obtained as follows.
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