Content deleted Content added
m →Object actuality > Specializations > In Smalltalk-80: Anchor added. |
→Representation by sets: New section added. |
||
Line 569:
</ol>
== Representation by sets ==
The eigenclass model can be interpreted by means of set theory.<ref name="ome-set-rep">
{{cite web | url=http://www.atalon.cz/om/object-membership/set-rep/ | title=Object Membership: Set-theoretic representation}}</ref>
<!-- -->
Objects are represented by sets so that object membership structure is induced by set structure.
The correspondence is best described via a generalization of the [[#canonical_structure|canonical structure]] such that only those features are axiomatized that are essential w.r.t. set representation.
=== {{anchor|essential_structure}} Essential structure of ϵ ===
An ''essential structure'' of ϵ is a structure
<span style="white-space:nowrap">''(<u>O</u>, .ec, ≤, <u>r</u>)''</span>
where
''<u>O</u>'' is the set of objects,
''.ec'' is the ''eigenclass map'' between objects,
''≤'' is the ''inheritance'' relation, and
''<u>r</u>'' is the ''inheritance root'', a distinguished object.
Objects that are not inheritance descendants of ''<u>r</u>'' are ''terminal(s)''.
The structure is subject to the following axioms:
<ol style="list-style-type:none; margin-left: 2ex;">
<li style="text-indent:-3ex; margin-left:3ex;">(e~1)
Inheritance, ≤, is a partial order.
<li>(e~2)
The eigenclass map, ''.ec'', is an order embedding w.r.t. ≤.
<li>(e~3)
Objects from eigenclass chains of terminals are minimal in ≤.
<li>(e~4)
Every eigenclass is a descendant of ''<u>r</u>''.
<li>(e~5)
The eigenclass chain of ''<u>r</u>''
(i.e. the [[#Reduced_helix|reduced helix]] ''<u>R</u>'') has no lower bound in ≤.
</ol>
Like with canonical structures, an essential structure is given by
<span style="white-space:nowrap">''(<u>O</u>,'' ϵ'')''</span>
where
<span style="white-space:nowrap">(ϵ) = (''.ec'') ○ (≤)</span>.
=== The embedding ===
Any essential structure of ϵ can be embedded into a set ''<u>V</u>'' that is formed as a cumulative hierarchy over a set of [[urelement|urelements]], ''<u>U</u>''.
Elements of ''<u>U</u>'' can be [[Pure set|pure sets]]
that are minimal both in
<span style="white-space:nowrap">''(<u>V</u>,'' ∈'')''</span> and
<span style="white-space:nowrap">''(<u>V</u>,'' ⊆'')''</span>
so that
they just behave ''like'' urelements with respect to ''<u>V</u>''.
<!-- -->
The set ''<u>V</u>'' is built in ''ω''+1 stages.
The 0-th stage is the set ''<u>U</u>'' of urelements.
For each natural number ''i'', the ''i''-th stage equals the ''i''-th application of ''P<sub style="margin-left:-.8ex">⋆</sub>'',
the ''powerset cumulation'' operator, defined by
:''P<sub style="margin-left:-.8ex">⋆</sub>(X)'' = (''P(X)'' ∖ {∅}) ∪ ''X''
where ''P(X)'' denotes the powerset of ''X''.
The ''ω''-th stage, called ''[[Universe_(mathematics)#In_ordinary_mathematics| superstructure]]'' in the field of [[non-standard analysis]],{{efn|
However, we provide a slightly different construction by removing the empty set.
}}
is the union of all previous stages.
This set stands for the inheritance root – it is therefore denoted ''<u>r</u>''.
The set ''<u>V</u>'' then equals the powerset cumulation of ''<u>r</u>'',
i.e. ''<u>V</u> = P<sub style="margin-left:-.8ex">⋆</sub><sup>ω+1</sup>(<u>U</u>) = P<sub style="margin-left:-.8ex">⋆</sub>(<u>r</u>)''.
The eigenclass map, ''.ec'', is defined on ''<u>V</u>'' by
:''x.ec = P(x)'' ∩ ''<u>r</u>'',
i.e. the eigenclass of ''x'' is the set of all those subsets of ''x'' which belong to some finite stage.
<!-- -->
Then
<span style="white-space:nowrap">''(<u>V</u>, .ec,'' ⊆'', <u>r</u>)''</span> is an essential structure of ϵ.
The inheritance relation, ≤, coincides with set inclusion, ⊆, on ''<u>V</u>''.
Object membership on ''<u>V</u>'' is given by:
''x'' ϵ ''y'' iff ''P(x)'' ∩ ''<u>r</u>'' is a subset of ''y''.
Terminal objects are the urelements.
The following are satisfied:
<!-- -->
<table border=0 cellpadding=2 style="margin:1ex 3ex;">
<tr valign=top>
<td>''x.ec'' = {''x''}</td>
<td>if ''x'' is terminal
(or ''x'' is from the eigenclass chain of a terminal),</td>
</tr>
<tr valign=top>
<td style="padding-right:2ex;white-space:nowrap;">''x.ec = P(x)'' ∖ {∅}</td>
<td>if (and only if) ''x'' ∈ ''<u>r</u>'',</td>
</tr>
<tr valign=top>
<td>''x.ec'' ⊂ ''x''</td>
<td>if (and only if) ''x'' ϵ ''x''.</td>
</tr>
</table>
Any subset ''<u>O</u>'' of ''<u>V</u>'' such that ''<u>r</u>'' ∈ ''<u>O</u>'' and ''<u>O</u>.ec'' = ''<u>V</u>.ec'' ∩ ''<u>O</u>'' forms an "object system": the substructure
<span style="white-space:nowrap">''(<u>O</u>, .ec,'' ⊆'', <u>r</u>)''</span> is an essential structure of object membership.
<!-- -->
Conversely, any essential structure of ϵ can be represented by such set ''<u>O</u>''.
Moreover, such a representation exists that
for every ''x'', ''y'' from ''<u>O</u>'',
:''x'' ∈ ''y'' iff ''x'' ϵ ''y'' and ''x'' ∈ ''<u>r</u>''.
== Specializations ==
| |||