Re: 'all resources' is not a set

>Another little suggestion for theoretical underpinnings
>of URIs: I suggest abandoning any formalism where
>'resources' form a set.
>
>Defining resources as a class rather than as
>a set has several advantages.

Do you mean 'proper class' as in some formal set theories? Or do you 
mean 'intensional class' as in some semantic theories? If the former 
I would strongly disagree with this idea. If the latter, I agree this 
is a neat idea (and RDF tries to accommodate it) but (1) it is highly 
controversial and(2) it does not require abandoning set theory as a 
mathematical device in describing the semantic foundations.


>With a class, there
>is no need for a well defined equality. The
>"same" relation does not have to be well defined
>for all pairs.

That is true for either reading. However, I do not see that as 
particularly as an advantage, particularly for a semantic 
foundational theory. On the contrary, that is a potential disaster.

>You get to eschew trying to have a well-defined notion
>of equality amongst resources. When asked 'can
>two different resources have the same URI', you can
>counter that the question is ill-formed, because there
>is no way, in general, to tell if two resources are
>the 'same' or 'different'. Is the referent of
>'the morning star' and of 'the evening star' the same?
>Not defined.

You chose a very bad example to make your case. The whole point of 
the example is that it IS well-defined: the answer is yes, they do 
have the same referent, viz. the planet Venus (but some people don't 
know that, hence the point of the example.)

>Now, there is, of course, the set of all URIs, and
>a (more or less) well defined equality relationship
>amongst URIs.
>
>There are still some axioms that can be stated without
>having equality. One axiom you may want is to assume
>that, whatever the 'meaning' is for a URI, it doesn't
>depend on time, the observer, or the state of the
>observer.

If you can't use equality on times and states then you have to 
explain what this claim is supposed to mean.

>But avoid treating resources as if they formed
>a set.

Nobody has yet defined what this damnably silly word "resources" is 
supposed to mean.  In the meantime, we do know quite a lot about 
things that form sets. It is very hard indeed to come up with an 
example of things that do not form a set. In fact it is provably 
impossible to *exhibit* such a thing: one has to resort to 
transfinite diagonalization arguments to conclude that there are any. 
As far as I know, these all depend one way or another on the ZF axiom 
of foundation, and it has been shown that one can get a perfectly 
fine set theory without assuming that axiom. So for the last decade 
or so, to even talk about things that do not form sets has been an 
act of mysticism.

To abandon set theory is basically to abandon the foundations of 
mathematics. That doesnt seem to me to be a very promising start 
towards constructing a useful foundation for anything.

Pat

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Received on Thursday, 25 September 2003 19:31:09 UTC