Eigenclass model: Difference between revisions

Revision as of 18:45, 23 April 2013

Revision as of 18:46, 23 April 2013

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=== The embedding ===

Any essential structure of &#1013; can be embedded into a set ''<u>V</u>'' that is formed as a cumulative hierarchy over a set of [[urelement|urelements]]s, ''<u>U</u>''.

Elements of ''<u>U</u>'' can be [[Pure set|pure sets]]

that are minimal both in

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:''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_Universe (mathematics)#In_ordinary_mathematicsIn 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 &ndash; 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>^ω+1(<u>U</u>) = P<sub style="margin-left:-.8ex">⋆</sub>(<u>r</u>)''.

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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|- valign=top>

</tr>

<td| style="padding-right:2ex;white-space:nowrap;"> | ''x.ec = P(x)'' ∖ {∅}</td>

<tr valign=top>

<td>| if (and only if) ''x'' ∈ ''<u>r</u>'',</td>

<td style="padding-right:2ex;white-space:nowrap;">''x.ec = P(x)'' ∖ {∅}</td>

<tr|- valign=top>

<td>if (and only if) ''x'' ∈ ''<u>r</u>'',</td>

<td>| ''x.ec'' &sub; ''x''</td>

</tr>

<td>| if (and only if) ''x'' &#1013; ''x''.</td>

<tr valign=top>

|}

<td>''x.ec'' &sub; ''x''</td>

<td>if (and only if) ''x'' &#1013; ''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