For Leibniz, the predicate "is equal to a" is true for b iff b satisfies all the predicates satisfied by a. How should one deal with the complications introduced by the ramified hierarchy?
For example, one obtains ramified analysis by associating an ordinal rank to set variables; roughly, in the definition of a set of a given rank, quantifiers range only over sets of lower rank, i.e., those whose definitions are logically prior.
The basic Kantian insight was further ramified by Lotze: mechanism and teleology are coexisting, compatible modes of explanation, each with their own proper sphere, and not contradictory opposites: "[b]oth … are limited, and both imply one another.
We iterate then ramified hierarchy up to this ordinal u(1) and get a system of ordinal strength u(2)> u(1), etc.
In this model the collection of all subsets of the set of natural numbers (represented by predicates) is of cardinality ω1 and is similar to the ramified hierarchy.
These conflicts underlie the "great debate" that broke out in French life science and quickly ramified into a popular public controversy in the late 1820s between Geoffroy St. Hilaire and Cuvier (Appel 1987).
But even the simple theory of types eventually fell out of favor as a foundational theory for mathematics possibly because in the wake of the Byzantine ramified theory, no version of type theory could find favor among mathematicians themselves.
This example illustrates well the complexities introduced by the ramified hierarchy.
Since arithmetic has for ordinal strength ε0 we consider first the iteration of the ramified hierarchy up to ε0.
This project has recently been dubbed "ramified" natural theology by Richard Swinburne (see Holder 2013).
Both authors rejected ramified type theory (RTT, in short) and the axiom of reducibility.
