User blog comment:Tetramur/My thoughts about functions and numbers/@comment-35470197-20191229115044/@comment-39541634-20200102054245

@Rpakr

"Machine X computes α" is shorthand for "Machine X computes an ordinal notation of α".

At the bear minimum, this means that Machine X is capable of:

(1) Deciding whether a given input is a valid representation of of some ordinal less α in that notaton.

(2) Comparing two valid strings and deciding which one of them represents a larger ordinal.

For example, if we have a TM whose alphabet is the symbol set {ω,+,*,^,,0}, and if that TM can:

(1) Detect whether an input string is in Cantor Normal Form.

(2) Given two Cantor Normal Forms, telling us which one is larger.

Then we could say that TM computes ε0.