Re: Is T an ancestor of T?

["Björn Persson" <spam-away@nowhere.nil>]

Georg Bauhaus wrote:
> Björn Persson <spam-away@nowhere.nil> wrote:
> : Georg Bauhaus wrote:
> : 
> :> By the last sentence of (10),
> :> the ultimate ancestor (T) "is not a descendant of any other type".
> :> So if T is the ultimate ancestor, it is not a descendant of T.
> :> Which it is by the first sentence of (10).
> :> I must be missing something. Specific types?
> : 
> : Maybe it's the word "other" you're missing?
> 
> I don't speak English.  Is there a mix of defined (linguistic)
> "ambiguity" (T descends from itself) and assumed clarity of "other"
> in this paragraph? (Is there a well defined mathematical meaning of
> "other" in general, not just in paragraph 3.4.1(10)? What is the set
> of "not any other types"?)

Oh come on! I'm sure German has the same construct. "Other" implies 
"other than X", and X has got to be mentioned or otherwise apparent from 
the context. The sentence is as follows:

"The ultimate ancestor of a type is the ancestor of the type that is not 
a descendant of any other type."

There are two types mentioned that are possible candidates for X. One, 
T1, is being defined as "the ultimate ancestor". The other, T2, is 
spoken of as "a type" and "the type". It is stated that T1 is an 
ancestor of T2, which means that T2 is a descendant of T1. It is also 
stated that T1 is not a descendant of any type except X. So which of T1 
and T2 is X?

If T1 and T2 are the same type, then they are both X, and the sentence 
says that the ultimate ancestor T of a type T is the ancestor T of the 
type T that is not a descendant of any type except T. No problem.

If T1 and T2 are not the same, and T1 is X, then it says that in order 
to be the ultimate ancestor, T1 must not be a descendant of any type 
except T1. That is, "other" means "other than itself".

If T1 and T2 are not the same, and T2 is X, then it says that in order 
to be the ultimate ancestor, T1 must not be a descendant of any type 
except T2. But T1 is an ancestor of T2 so it can't be a descendant of 
T2, and then T1 isn't allowed to be a descendant of any type at all. Not 
  only does this cause a contradiction, but it's also terribly 
convoluted. Why would anyone write "any other type" if "any type" would 
mean the same thing? Obviously they don't mean that T2 is X.

> : By the way, did you notice that class-wide types aren't ancestors or 
> : descendants of themselves?
> 
> Yes, and class-wide types don't have parents or ancestors at all,
> do they?

Yes they have. T is an ancestor of T'Class. I'd assume it's the parent 
too, but I haven't found a proof for that.

-- 
Björn Persson                              PGP key A88682FD
                    omb jor ers @sv ge.
                    r o.b n.p son eri nu