Re: FW: data structure vs. mathematical structure (was: [Axiom-developer] Graph theory)

[Gabriel Dos Reis]

"Bill Page" <address@hidden> writes:

| > Bill Page writes:
| > | > | All domains that have SetCategory are required to have a
| > | > | hash| into SmallInteger.
| > | >
| > | Gaby wrote:
| > | > That is not a mathematical requirement.
| > |
| > Bill Page wrote; 
| > | I would tend to agree but perhaps Kurt Gödel would not have...
| > | ;-)
| > 
| 
| On November 14, 2006 12:01 AM Gaby wrote:
| 
| > From constructive mathematics point of view, the only things
| > that are required for a set are:
| > 
| >   (1) say how to build element of a set
| >   (2) equality test.
| > 
| 
| No, there is a lot more to the mathematics of set than that. It
| would mean that all sets are finite and that is quite far from
| the case.

How do you arrive to that conclusion?

| Rather, sets should be strongly related to types.
| 
| I think the "constructive mathematics" that is most suitable
| to Axiom is probably is probably Intuitionist type theory
| (Martin-Löf). See:

In fact, I prefer the definition given by Erret Bishop.  See chapters 1
and 2 "his" book

   "Constructive Analysis", 

       Erret Bishop
       Douglas Bridges


>From my perspective, it shows a much deeper impact.

-- Gaby