On Mon, Jan 30, 2006 at 09:47:24AM -0000, West, Matthew R SIPC-DFD/321 wrote:
> [Pat C wrote:]
> > The other essential characteristic of a Type is that it is a
> > grouping (whatever one wants to call it), and has a cardinality,
> > which can be zero. Of course, we may not know the cardinality and
> > it can change over time, but in any given context (e.g. time
> > interval - assuming time is relevant to the definition of a
> > particular Type and its instances), it will have some cardinality.
>
> MW: Again not problematic. (01)
Not not problematic! :-) At least, there are land mines in certain
areas where you have to tread carefully. Notably: In most modern set
theories -- including ZF -- there are, in a precise sense, too many sets
for the lot of them jointly to have a cardinality. Hence, if we have a
type SET, it will not have a cardinality. (02)
> > An instance of a Type is not the same as a member of a mathematical
> > set. (03)
More exactly, I think: the instance_of relation between things and types
is a different relation from the membership relation between things and
sets. (04)
> > However, the cardinality of a Type is the same as the number of its
> > instances, which are **not** set-theoretic members of the Type. (05)
I have no idea what that last bit means. What exactly are you denying
here, Pat? (06)
> > There is no first-order distinction between classes and unary
> > relations. (07)
Whether there is or there isn't has nothing to do with order. It simply
depends on whether we want to make it so or not, e.g., by asserting that
classes are extensional and unary relations aren't in our axioms. As
for issues of first-order vs higher-order, the theory that is ever so
slowly emerging here seems to be counting everything as a first-class
citizen, classes and relations included, so the framework appears to be
fully first-order. Just because you are quantifying over things that
you call, and that behave like, classes and relations doesn't mean you
aren't first-order. (08)
> > One is free to define a second-order predicate that makes the
> > distinction. (09)
There is nothing about such a predicate that makes it inherently
second-order. Whether or not it is depends on the semantics of your
language. (010)
> MW: I'm not trying to insist that they are sets for you. Only that
> they can be for me. (011)
Seems to me that's easy enough (modulo the possible problems of
cardinality and such noted above), so long as you are silent in the base
axioms on the question of whether or not types are extensional. That
leaves it open for someone to add an extensionality axiom. (012)
> > Now, if we must, *must*, **must** define a "Type" as a type of
> > mathematical set, according to the above interpretation, then "Type"
> > would be a subset of "Set" as described by the Frame-Ontology. (013)
Dicey. And we really can't say one way or another until we have axioms
for both. (014)
-chris (015)
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