Dear Chris, (01)
Well let's start by confusing the situation with a few facts. (02)
>
> Barry's criticisms of the use of a non-well-founded set theory like
Aczel's
> AFA are on the money. (03)
MW: ISO 15926 does say that a class can be a member of itself for reasons
Chris P has alluded to, but it is not committed to a particular form of
non-well foundedness. Aczel is mentioned in an informative note, and there
is an informative annex that gives a lay explanation of some different sorts
of structures sets might have, including well-founded and non-well founded
sets. (04)
> He notes that it is a greatly overpowered for the needs
> of the document; it entails, among other things, the entire massively
infinite
> hierarchy of transfinite numbers. (How massive? So massive that there
is no
> transfinite number big enough to number them.) (05)
MW: I don't know why this should be a problem, since there is no commitment
to record them all. (06)
> Moreover, ironically, AFA and
> its like are in a sense underpowered as well for the given task. Notably,
as I
> understand the document, THING is itself a class that contains, well,
> everything. (07)
MW: Yes. Every upper ontology has such an class, so ISO15926 is not the only
one with this problem. (08)
> The existence of such a class (understood as a non-wf set) is
> flatly inconsistent with non-wf ZF spinoffs like AFA. (09)
MW: I had missed that. Could you give me a section reference (I have Aczel's
book).
>
> Bottom line (as John Sowa likes to say): The underlying class theory of
the
> document needs to be thrown out and rethought completely. (010)
MW: What would you suggest? Not much point kicking up a storm unless you
have a better alternative. (011)
Regards (012)
Matthew West
Information Junction
Tel: +44 560 302 3685
Mobile: +44 750 3385279
matthew.west@xxxxxxxxxxxxxxxxxxxxxxxxx
http://www.informationjunction.co.uk/
http://www.matthew-west.org.uk/ (013)
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