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[Axiom-developer] Re: FW: data structure vs. mathematical structure

From: Ralf Hemmecke
Subject: [Axiom-developer] Re: FW: data structure vs. mathematical structure
Date: Tue, 14 Nov 2006 23:03:26 +0100
User-agent: Thunderbird (X11/20061025)

On 11/14/2006 10:17 PM, Page, Bill wrote:
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.
| > Bill Page wrote: | 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.

On Tuesday, November 14, 2006 1:20 PM Gaby wrote:
How do you arrive to that conclusion?

I thought I was stating something obvious.

I remember I said that "Set" is somehow a bad name for a domain in Axiom that only implements "(the collection of) finite sets of elements of a given type T".

Or (from sets.spad):

++ A set over a domain D models the usual mathematical notion of a
++ finite set of elements from D.


i: Integer


s: FinitePowerSet T

would be in perfect analogy if one read ":" as "element of", then to go on "l: List T" would mean "List" is the container of all finite sequences (with some information about their representation (linked list)). It's soon getting confusing. So I would rather choose "FiniteSet". But then (except proper classes) everything is a set. Why would one need a domain of sets? "SetCategory" is more important.

And in Axiom it is an approximation anyway, since it is Set(T), ie a collection of things of a common type T. The name "Set" is probably an exception one could accept.


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