On Jan 15, 2010, at 5:13 PM, John F. Sowa wrote: (01)
> Pat,
>
> 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.
>
> 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.
>
> But I just wanted to note that infinite sets (and extremely large
> finite sets) can only be specified in two ways:
>
> 1. By some rule or rules that generate them (e.g., a grammar).
>
> 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).
>
> But case #2 depends on case #1, since the integers themselves
> must be specified by some rules or axioms. (02)
Most mathematicians would disagree. The natural numbers cannot be
fully specified by any finite number of axioms, yet we all feel that
we know what they are, and are quite happy to refer to them. (03)
>
> PH> Neither view is 'correct', and there is no fact of the matter.
>
> I certainly agree.
>
> 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.
>
> 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.
> (04)
They are not going to be displayed in their full extension, of course.
But they can be specified and mentioned and thought about. We will
almost certainly need these sets in this very project: the set of all
points in 3-d space; the set of all connected subsets of points in 3-d
space; the set of all timepoints; the sets of all natural numbers and
of all rational numbers; and the set of all measuring scales of
various types. All of these sets are infinite, some of them *very*
infinite. (05)
Pat (06)
> John
>
>
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> (07)
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