Pat, (01)
PH> As I pointed out in an earlier message in this thread, what
> one thinks of as natural or intuitive here depends largely
> on which flavor of foundations of mathematics one prefers.
> I myself was raised in a set-theoretic framework, in which
> every mathematical construct is understood in the final analysis
> to be a set. (02)
I have no objection to the extentional view, and for many purposes
it is very convenient. The easiest way to prove that two rules
are equivalent can often be to consider the sets they generate. (03)
But I just wanted to note that infinite sets (and extremely large
finite sets) can only be specified in two ways: (04)
1. By some rule or rules that generate them (e.g., a grammar). (05)
2. Or by a function applied to some other infinite set (e.g.,
the times-two function for mapping the integers to the
even integers). (06)
But case #2 depends on case #1, since the integers themselves
must be specified by some rules or axioms. (07)
PH> Neither view is 'correct', and there is no fact of the matter. (08)
I certainly agree. (09)
PH> However, that said, CL semantics is rooted in set theory
> rather than category theory. So in CL, relational extensions
> are indeed sets of tuples. (010)
I have no desire to change or redo the CL semantics. But it
is worth noting that when CL is being used to express theories
with infinite models, those sets might "exist" in some Platonic
heaven, but they aren't going to specified, listed, stored, or
exhibited in any physical medium. (011)
John (012)
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