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<a name="A-Hierarchy-of-Sets"></a>
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<a name="A-Hierarchy-of-Sets-1"></a>
<h4 class="subsubsection">2.7.15.1 A Hierarchy of Sets</h4>

<p>As indicated already, the virtual machine represents all functions and
data as members of a set satisfying the properties in <a href="Raw-Material.html#Raw-Material">Raw Material</a>,
namely a <code>nil</code> element and a <code>cons</code> operator for constructing
trees or nested pairs of <code>nil</code>. 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.
</p>
<p>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.
</p>
<p>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 &ldquo;trees&rdquo; is reserved for members
<a name="index-trees-3"></a>
of the bottom set, as in &ldquo;for any tree <code><var>x</var></code> &hellip;&rdquo;.
Unless otherwise stated, variables like <code><var>x</var></code> and
<code><var>y</var></code> are universally quantified over the bottom set only.
<a name="index-universal-quantification-1"></a>
</p>
<p>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 <code><var>x</var>_<var>n</var></code> denote the image in
the <code><var>n</var></code>th level set of a tree <code><var>x</var></code> in the bottom
set. The level numbers in this notation start with zero, and we take
<code><var>x</var>_0</code> to be synonymous with <code><var>x</var></code>. For good measure,
let <code>(<var>x</var>_<var>n</var>)_<var>m</var></code> = <code><var>x</var>_(<var>n</var>+<var>m</var>)</code>.
</p>
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