Chapter 14

Strong Normalisation for F

The aim of this chapter is to prove:

Theorem All terms of F are strongly normalisable, and the normal form is unique.

The uniqueness is not problematic: it comes from an extension of the Church-Rosser theorem. Existence is much more delicate; in fact, we shall see in chapter 15 that the normalisation theorem for F implies the consistency of second order arithmetic PA2. The classic result of logic, if anything deserves that name, is G•odel's second incompleteness theorem, which says (assuming that it is not contradictory) that the consistency of PA2 cannot be proved within PA2. Consequently, since consistency can be deduced from normalisation within PA2, the normalisation theorem cannot be proved within PA2. That gives us an essential piece of information for the proof: we must look for a strategy which goes outside

PA2.

Essentially, PA2 contains the Axiom (scheme) of comprehension

9X: 8 : ( 2 X , A[ ])

where A is a formula in which the variable X does not occur free. A may containrst order (8 : , 9 : ) and second order (8X: , 9X: ) quanti cation. Intuitively, the rst order variables range over integers and the second order ones over sets of integers. This system su ces for everyday mathematics: for instance, real numbers may be coded as sets of integers.

So we seek to use \all possible" axioms of comprehension, or at least a large class of them. For this, we shall look back at Tait's proof (using reducibility) and try to extend it to system F.

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