Curiously, if you replace “finite” with “countable” and “infinite” with “uncountable” then it is possible to devise such an encoding scheme. (This is connected with the existence of “Aronszajn trees”.)
Suppose we have a countable alphabet of symbols A, and we want to encode countable ordinals using functions alpha → A for countable ordinals alpha, then there exists an encoding scheme such that, after only countably many symbols have been received, you can either say “yes, this represents the countable ordinal x” or “no, this isn’t a valid encoding”.
(Consider the set of all increasing functions from alpha into the rationals, where alpha is a finite or countable ordinal.)
Curiously, if you replace “finite” with “countable” and “infinite” with “uncountable” then it is possible to devise such an encoding scheme. (This is connected with the existence of “Aronszajn trees”.)
Suppose we have a countable alphabet of symbols A, and we want to encode countable ordinals using functions alpha → A for countable ordinals alpha, then there exists an encoding scheme such that, after only countably many symbols have been received, you can either say “yes, this represents the countable ordinal x” or “no, this isn’t a valid encoding”.
(Consider the set of all increasing functions from alpha into the rationals, where alpha is a finite or countable ordinal.)