  In this article we want to see that it is possible to iterate and generalize
the notions presented in "Carlson's <_1-relation on the class of epsilon
numbers" such that we can obtain the higher or thinner classes of ordinals
induced by the <_1-relation as solutions of conditions. Indeed, these results
are so general that in a future article we will be able to characterize the
ordinal |Pi_1^1 - CA_0 | as the smallest ordinal of the thinnest class induced
by <_1.
