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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