Re: SUO: Echos
Chris,
That is a good summary of the issues concerning sets,
which highlights the notion of "rank" as central.
And it is related to the reason why I like the iterative
construction for domains of discourse in metalanguages.
>Reflection on Zermelo's original axioms and subsequent
>work in set theory eventually led to the development (by Mirimanoff, von
>Neumann, and Zermelo himself, among others) of the so-called "iterative"
>conception of set on which sets fall into a natural hierarchy based upon
>the membership relation: start with some "urelements" (concrete objects,
>say), then the first level consists of sets of urelements, and the n+1th
>level consists of all the sets that can be formed out of the objects in
>the first n levels together with the union of those levels. Unions are
>taken at limit stages. The members of earlier levels thus accumulate in
>later levels, and so this conception is often also called the
>"cumulative hierarchy". It is the "intended" model of ZF for most
>working set theorists.
>Now, say that the *rank* of a set is the ordinal number that indexes the
>level of the hierarchy in which that set first appears. Then we can
>draw the distinction between sets and proper classes simply and cleanly
>as follows: sets are those collections that have a rank. Proper
>classes, by contrast, are collections that contain sets of arbitrarily
>high rank; their members occur arbitrarily high up in the hierarchy.
>There is thus no level of the hierarchy at which a proper class ever
>appears, no level at which it is "formed" from objects in lower levels,
>and hence there are no collections that are themselves formed out of
>*them* -- i.e., unlike sets, they can't be members of other classes.
This is also why I like to use the iterative (or cumulative
construction) for domains of discourse in languages with
multiple metalevels. There is a domain D0 at the bottom,
and every domain Di+1 consists of the union of Di with the
set of syntactic constructs of the vocabulary of language Li.
Then if you have a language without such an obvious iterative
construction, such as English, you can say that sentences that
cannot be mapped into an iterative form are not well-formed
for the purpose of being interpreted in the metalevel hierarchy.
For example, the sentence "The last sentence in your previous
quotation is true or false" would be well-formed because
it could be mapped into a construction that would give it
a finite rank. But the typical "liar" sentence:
The Cretan said "Everything I say is false".
is not well-formed because it cannot be assigned a finite rank.
John Sowa