> 1) Domain is the domain whose values are domains.
>
> Is this ok?
This says that Domain contains itself as a value.
> 2) A category is some specified subdomain (subset) of Domain?
This doesn't explain what a category is, but only that all categories
are contained in Domain and thus are called 'domain's. Unfortunately,
that "specific subdomain of Domain" could be empty, so I don't know
whether there are categories at all. ;-)
> 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).
I think comparing subdomain with subset makes it clear enough for me.
> 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.
Don't you mean to an "element of the power set of Domain", i.e. Categoy
is a subdomain/subset of Domain?
> 5) Type is the category in Category corresponding to Domain. In other
> words Type is Domain in Category.
That sound like "Domain: Type". But we also have "Type: Category" and
"Category \subdomain Domain". That says that "Type: Domain" is also OK.
That looks all too circluar to me.
> 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.
But Saul defines
Domain: Category == with
so that is yet another occurence of "Domain". Would you say that in
Aldor this Youssef-"Domain" is what "Type" is in panAxiom?
Ralf
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