As we have seen in Chapter 10, the main property of typed systems not possessed by untyped systems is that all reductions are finite, and hence every typed term has a normal form. In this appendix we shall prove this theorem for the simply typed systems in Chapter 10, and for an extended system from which the consistency of first-order arithmetic can be deduced.

The proofs will be variations on a method due to W. Tait, [Tai67]. (See also [TS00, Sections 6.8, 6.12.2] or [SU06, Sections 5.3.2–5.3.6].) Simpler methods are known for pure λ and CL, but Tait's is the easiest to extend to more complex type-systems.

We begin with two definitions which have meaning for any reduction concept defined by sequences of replacements. The first is a repetition of Definition 10.14. The second is the key to Tait's method.

Definition A3.1 (Normalizable terms) A typed or untyped CL-or λ-term X is called normalizable or weakly normalizable or WN with respect to a given reduction concept, iff it reduces to a normal form. It is called strongly normalizable (SN) iff all reductions starting at X are finite.

As noted in Chapter 10, SN implies WN. Also the concept of SN involves the distinction between finite and infinite reductions, whereas WN does not, so SN is a fundamentally more complex concept than WN.