Re: set incompleteness

```John,

On 04 Jul 2009, at 18:24, John Mikes wrote:```
```
> Dear Bruno, I mentioned that I have something more on the 'set' as
> you (and all since G. Cantor) included it in the formulations. I had
> a similar notion about my "aris-total", the definition of Aristotle
> that the 'total' is always more than the 'sum' of its components. Of
> course, at the time when A. thought about it, 'components' were only
> 'physical objects' included in an ensemble as individual and
> unrelated noumena.

We will see how we can do something similar with set. few
mathematicians are really interested in sets, but in sets together
with a structure (usually determined by operations and relations on
the set.

>
> If you advance in our epistemic cognitive inventory to a bit better
> level (say: to where we are now?) you will add (consider) relations
> (unlimited) to the names of 'things' and the increased notion will
> exactly match the 'total' (what A was missing from the 'sum'). It
> will also introduce some uncertainty into the concept (values?) of a
> set.

I am not sure that I understand.

> I see a similar situation with your ways writing of
> 'sets' (1,2,3...) - or: ( 1, 2, 3... )

I guess you mean {1, 2, 3 ... }. "{" and "}" are standard, and "(" and
")" will be reserved for other things, like delimiter of expression,
like in (3+4), or the notion of couples (soon to be introduced).

> neglecting the additional relations maybe expressed in the
> (neglected) commas, spaces, even the parentheses. All may mean
> something and that meaning gives completeness to the entire set
> beyond the 'factual' elements 1 2 3 . I don't know 'what', but for
> sure something well pertinent. In infinite sets such uncertainty may
> amount to infinite uncertainty.

I don't see anything uncertain in most infinite sets. But this will be
scrutinized soon, or a bit later ... Some sets will appear more
complex than other, and *some* set will have "uncertainties" attached
to them, but to understand this we have to progress a bit more.

Bruno

http://iridia.ulb.ac.be/~marchal/

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