On Tue, Jul 8, 2008 at 6:26 PM, Ralf Hemmecke wrote:
> Bill,
>
> my knowledge of the English language doesn't let me make sense out of this
> sentence:
>
> "Type is the category in Category consisting of all domains in Domain."
>
> on
> http://axiom-wiki.newsynthesis.org/[EMAIL PROTECTED]
>
> Could you make it simpler?
>
Hmmm.... It's hard to make it simpler. How about if I make it longer,
then each part might be simpler. :-)
Starting in reverse:
1) Domain is the domain whose values are domains.
Is this ok?
2) A category is some specified subdomain (subset) of Domain?
I.e. The subset of Domain corresponding to a category is given by the
domains which refer to specific categories in their definition.
3) A subdomain need not be a proper subset, i.e. it can consist of all
of Domain as a whole.
In set theory X is an element of the power set power(X).
4) Category is the domain of categories, i.e. subdomains of Domain. As
such it corresponds to the concept of the power set of Domain.
5) Type is the category in Category corresponding to Domain. In other
words Type is Domain in Category.
Just like X in power(X).
> Somehow I now have the impression that Aldor-Type and OpenAxiom-Type
> are different.
Yes. According to my definition they are different.
> That "Type" is a category in OpenAxiom would have never come to
> my mind in Aldor.
>
Type in OpenAxiom is like 'with {}' in Aldor. Until Saul Yousseph
presented his ideas on implementing mathematical category theory in
Aldor perhaps no one thought this category was particularly
interesting.
Regards,
Bill Page.
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