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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 1.5.0.8 (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".
```

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

Although

i: Integer

and

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.
```
Ralf

```

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