> (That said, I don't understand why this discussion is relevant at all.
> The fact that the ordering exists doesn't mean that one would want to
> declare the Ord instance, like with complex numbers.)

It's not relevant, it's just an intellectual exercise.

For any set with a equality relation there exists a total order relation on that set consistent with the equality relation. The question is then whether there exists a set with a computable equality relation such that there is no computable total order.

I think the following computable function shows that it is always possible (it chooses an order during queries):

Maintain a table, initially emtpy

As soon as (a <= b) is requested, see if a and b are already in the table (using the computatble equality function) , if so, use their ordering in the table.
If an element is not in the table, add it. 

Hence the table gives a consistent total order (it depends on which ordering queries are requested, but that is not relevant?)






2015-01-01 16:06 GMT+01:00 Atze van der Ploeg <atzeus@gmail.com>:

Nope you're right. Indeed uncompatible with the field structure. Now I'm confused :)

I now understand your question, but do not immediately know the answer. Anyone?

On Jan 1, 2015 4:02 PM, "Tom Ellis" <tom-lists-haskell-cafe-2013@jaguarpaw.co.uk> wrote:
On Thu, Jan 01, 2015 at 03:52:26PM +0100, Atze van der Ploeg wrote:
> If we do not require that (a <= b) && (a >= b) ==> a == b (where <= is
> from the total ordering and == is from the equality relation) then it is
> trivial, take the total ordering forall x y.  x <= y that i mentioned
> earlier.
>
> So the compatiblity with equality (you say field structure) is not besides
> the point, in fact antisymmetry means that the ordering corresponds to the
> equality relation.
>
> Clear now or did I misunderstand?

Here is my proposed equality and ordering on the complex numbers:

    data Complex = Complex (Double, Double) deriving (Eq, Ord)

Does this violate any of my requested conditions?
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