Let [ ] : Ord -> Relation Nat
be a function that realizes an ordinal into a well-ordering of the naturals with that order type.
Given a relation R on the naturals, we define the restricted relation R(<n) to be R but just acting on the first n naturals.
Defn: [ ]
is continuous iff for every series of ordinals a_i
whose limit is l
, for every natural n
, there exists an i
such that for all j > i
, [a_i](<n) = [l](<n)
.
Intuitively, if you are only looking at a finite prefix of the naturals, you can get to a limit ordinal by going finitely high up its limit sequence.