> There is not one Dedekind-complete ordered field
Yes there is.
> For example the nxn diagonal matrices with real entries
That's a model, not the theory.
> I don't know why you are conflating the words "compute" and "define"
This whole discussion is about definition.
Your own comment started with "You first define..."
And no, you are not able to compute arbitrary products of arbitrary reals in this way anyway, if by computation you mean a finitary algorithmic process.
Ok, I think your definition of "field" is different to mine. Mine starts "A field is a set F with binary operations + and x" and then goes on to list some properties they have to have, yours seems to be doing something different. If you start with mine you get a whole bunch of different complete ordered fields, but you can easily show they're all isomorphic (e.g. Spivak does this IIRC).
>This whole discussion is about definition
No, it isn't this whole discussion is answering the question about computing pi * pi. Maybe I was slightly sloppy in my first answer to that question, I was only attempting to sketch the method.
I don't really want to compute products of arbitrary reals in that way (or any way), but the method I sketched works for computable reals which is sufficient to cover the case of \pi. By computation I mean something that runs on a Turing machine, but I don't assume finitary (I'm happy for my Turing machine to keep producing more and more bits of precision forever).
Yes there is.
> For example the nxn diagonal matrices with real entries
That's a model, not the theory.
> I don't know why you are conflating the words "compute" and "define"
This whole discussion is about definition.
Your own comment started with "You first define..."
And no, you are not able to compute arbitrary products of arbitrary reals in this way anyway, if by computation you mean a finitary algorithmic process.