On Jan 15, 2010, at 7:25 AM, John F. Sowa wrote: (01)
> Pat,
>
> JFS>> I would prefer to say that a function is a mapping
>>> between two sets.
>
> PH> But then we have to say what a mapping is...
>
> In set theory, you specify the mapping extensionally by pairs
> or intensionally by rules or axioms.
>
> But in category theory ... (02)
Yes, I know. 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. Many people
prefer the category-theoretic approach, in which categories and
morphisms are taken as fundamental and set theory is simply one,
perhaps rather basic, category. Neither view is 'correct', and there
is no fact of the matter. However, that said, CL semantics is rooted
in set theory rather than category theory. So in CL, relational
extensions are indeed sets of tuples. (03)
>
> JFS>> For infinite sets, all specifications must be intensional.
>
> PH> What you say here could be interpreted as a claim that
>> infinite sets cannot be specified (since sets are extensional
>> and extensional must be finite?) which is of course not true.
>
> A specification of an infinite set by finite rules: (04)
Yes, of course it can be done. My point was only to note that your
previous message could be read as implying that it could not be done. (05)
Pat (06)
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