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As indicated already, the virtual machine represents all functions and
data as members of a set satisfying the properties in Raw Material,
namely a nil
element and a cons
operator for constructing
trees or nested pairs of nil
. However, it will be necessary to
distinguish the results of computations that go wrong for exceptional
reasons from normal results. Because any tree in the set could conceivably
represent a normal result, we need to go outside the set to find an
unambiguous representation of exceptional results.
Because there may be many possible exceptional conditions, it will be helpful to have a large set of possible ways to encode them, and in fact there is no need to refrain from choosing a countably infinite set. Furthermore, it will be useful to distinguish between different levels of severity among exceptional conditions, so for this purpose a countably infinite hierarchy of mutually disjoint sets is used.
In order to build on the theory already developed, the set that has been
used up to this point will form the bottom level of the hierarchy, and
its members will represent normal computational results. The members of
sets on the higher levels in the hierarchy represent exceptional
results. To avoid ambiguity, the term “trees” is reserved for members
of the bottom set, as in “for any tree x
…”.
Unless otherwise stated, variables like x
and
y
are universally quantified over the bottom set only.
Because each set in the hierarchy is countably infinite, it is
isomorphic to the bottom set. With respect to an arbitrary but fixed
bijection between them, let x_n
denote the image in
the n
th level set of a tree x
in the bottom
set. The level numbers in this notation start with zero, and we take
x_0
to be synonymous with x
. For good measure,
let (x_n)_m
= x_(n+m)
.
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