Editing Principia Mathematica (section)

==Ramified types and the axiom of reducibility==
<!--'Ramified Theory of Types', 'Ramified Type Theory', 'Ramified theory of types', 'Ramified type theory' redirect here-->
In simple type theory objects are elements of various disjoint "types". Types are implicitly built up as follows. If τ<sub>1</sub>,...,τ<sub>''m''</sub> are types then there is a type (τ<sub>1</sub>,...,τ<sub>''m''</sub>) that can be thought of as the class of propositional functions of τ<sub>1</sub>,...,τ<sub>''m''</sub> (which in set theory is essentially the set of subsets of τ<sub>1</sub>×...×τ<sub>''m''</sub>). In particular there is a type () of propositions, and there may be a type ι (iota) of "individuals" from which other types are built. Russell and Whitehead's notation for building up types from other types is rather cumbersome, and the notation here is due to [[Alonzo Church|Church]].

In the '''ramified type theory'''  of PM all objects are elements of various disjoint ramified types. Ramified types are implicitly built up as follows. If τ<sub>1</sub>,...,τ<sub>''m''</sub>,σ<sub>1</sub>,...,σ<sub>''n''</sub> are ramified types then as in simple type theory there is a type (τ<sub>1</sub>,...,τ<sub>''m''</sub>,σ<sub>1</sub>,...,σ<sub>''n''</sub>) of "predicative" propositional functions of τ<sub>1</sub>,...,τ<sub>''m''</sub>,σ<sub>1</sub>,...,σ<sub>''n''</sub>. However, there are also ramified types (τ<sub>1</sub>,...,τ<sub>''m''</sub>|σ<sub>1</sub>,...,σ<sub>''n''</sub>) that can be thought of as the classes of propositional functions of τ<sub>1</sub>,...τ<sub>''m''</sub> obtained from propositional functions of type (τ<sub>1</sub>,...,τ<sub>''m''</sub>,σ<sub>1</sub>,...,σ<sub>''n''</sub>) by quantifying over σ<sub>1</sub>,...,σ<sub>''n''</sub>. When ''n''=0 (so there are no σs) these propositional functions are called predicative functions or matrices. This can be confusing because modern mathematical practice does not distinguish between predicative and non-predicative functions, and in any case PM never defines exactly what a "predicative function" actually is: this is taken as a primitive notion.

Russell and Whitehead found it impossible to develop mathematics while maintaining the difference between predicative and non-predicative functions, so they introduced the [[axiom of reducibility]], saying that for every non-predicative function there is a predicative function taking the same values. In practice this axiom essentially means that the elements of type (τ<sub>1</sub>,...,τ<sub>''m''</sub>|σ<sub>1</sub>,...,σ<sub>''n''</sub>) can be identified with the elements of type (τ<sub>1</sub>,...,τ<sub>''m''</sub>), which causes the hierarchy of ramified types to collapse down to simple type theory. (Strictly speaking, PM allows two propositional functions to be different even if they take the same values on all arguments; this differs from modern mathematical practice where one normally identifies two such functions.)

In [[Ernst Zermelo|Zermelo]] set theory one can model the ramified type theory of PM as follows. One picks a set ι to be the type of individuals. For example, ι might be the set of natural numbers, or the set of atoms (in a set theory with atoms) or any other set one is interested in. Then if τ<sub>1</sub>,...,τ<sub>''m''</sub> are types, the type (τ<sub>1</sub>,...,τ<sub>''m''</sub>) is the power set of the product τ<sub>1</sub>×...×τ<sub>''m''</sub>, which can also be thought of informally as the set of (propositional predicative) functions from this product to a 2-element set {true,false}. The ramified type (τ<sub>1</sub>,...,τ<sub>''m''</sub>|σ<sub>1</sub>,...,σ<sub>''n''</sub>) can be modeled
as the product of the type (τ<sub>1</sub>,...,τ<sub>''m''</sub>,σ<sub>1</sub>,...,σ<sub>''n''</sub>) with the set of sequences of ''n'' quantifiers (∀ or ∃) indicating which quantifier should be applied to each variable σ<sub>''i''</sub>. (One can vary this slightly by allowing the σs to be quantified in any order, or allowing them to occur before some of the τs, but this makes little difference except to the bookkeeping.)

The introduction to the second edition cautions:
<blockquote>One point in regard to which improvement is obviously desirable is the axiom of reducibility ... . This axiom has a purely pragmatic justification ... but it is clearly not the sort of axiom with which we can rest content. On this subject, however, it cannot be said that a satisfactory solution is yet obtainable. Dr [[Leon Chwistek]] [Theory of Constructive Types] took the heroic course of dispensing with the axiom without adopting any substitute; from his work it is clear that this course compels us to sacrifice a great deal of ordinary mathematics. There is another course, recommended by Wittgenstein† (†Tractatus Logico-Philosophicus, *5.54ff) for philosophical reasons. This is to assume that functions of propositions are always truth-functions, and that a function can only occur in a proposition through its values. (...) [Working through the consequences] ... the theory of inductive cardinals and ordinals survives; but it seems that the theory of infinite Dedekindian and well-ordered series largely collapses, so that irrationals, and real numbers generally, can no longer be adequately dealt with. Also Cantor's proof that 2n > n breaks down unless n is finite.<ref>PM 1927:xiv</ref></blockquote>
<blockquote>
It might be possible to sacrifice infinite well-ordered series to logical rigour, but the theory of real numbers is an integral part of ordinary mathematics, and can hardly be the subject of reasonable doubt. We are therefore justified (sic) in supposing that some logical axioms which is true will justify it. The axiom required may be more restricted than the axiom of reducibility, but if so, it remains to be discovered.<ref>PM 1927:xlv</ref>
</blockquote>