**Chapters 8-12**, manuscript, 1982.

This second part expounds the theory of dilators and related topics, e.g., β-proofs. The book (written in 1982) was originally intended as a whole; Per Martin-Löf convinced me to split it. The sole chapters 1-7 were published in 1987. Several changes occurred on the side of the publisher which delayed the handling of part II. On my side, I had lost faith in infinitary logic and made no effort to proofread the crude LaTeX file of chapters 8-12. I issue the document as it is, poorly formated, e.g., with the arrows missing in diagrams. Although I think that the general orientation is wrong, it contains many ideas that can perhaps be reused in another context.

Typically, the idea of a *functor preserving direct limits and pullbacks,* the leitmotiv of this part II, was
reused to develop my own denotational semantics, *coherent spaces*, which would eventually lead to linear logic.

The original idea was to give a more satisfactory meaning to *ordinal notations*, the Cantor Normal Form and
its various generalisations. The basic remark is that the replacement of the ordinal ω with the integer n in
a notation yields a natural number, e.g., ω^{2} yields n^{2}. In other terms,
g_{ω2}(n)=n^{2}, g_{α} being the *slow growing
hierarchy*. This process of replacement can be made functorial, i.e., apply to morphisms - strictly monotonous
functions from m to n; such a functor preserves pull-backs and can be directly extended by means of direct limits:
this a *dilator*. This extension corresponds to the replacement of ω with arbitrary values, e.g.,
ω+1 or the first non denumerable ordinal Ω, in which case ω^{2} becomes (ω+1)^{2} or Ω^{2}.
The starting point of the approach was my proof (1975) of the *theorem of comparison of hierachies*. Let h be
your favourite fast growing hierarchy of functions from integers to integers; then h can be expanded into a
hierarchy H of functions from denumerable ordinals to themselves. Now, we can relate h (indexed with usual ordinal
notations) to H (indexed with the same, in which ω has been replaced with Ω), e.g. h_{γ}
vs. H_{Γ}. The two basically agree, i.e.,
h_{γ}(n)=g_{HΓ(ω)}(n). In particular, the fast growing hierachy up to
γ=ε_{0} is matched by the slow growing one up to the Howard ordinal H_{Γ}(0),
where Γ is the first ε number after Ω.