Re: VNSA-L digest 326

[Aiviet Nguyen <aiviet@cat.syr.edu>]

Hi Anh Vu,

   I guess that Sonnet meant "closed" as "close and open" in terms of 
topology. There is a theorem saying that an algebraic field if ordered
( if you can defined an order relation like >, =, <) and closed ( if you 
can define a topology as a system of open sets and closed sets to do the 
limit...) then it is isomorphic ( equivalent in a mathematically 
rigorous sense) to the set of real numbers.
   I think Sonnet used the phrase " in the real sense" as "like real numbers
case" or " for instance, real numbers", but not the vague meaning of daily 
life " in fact" or "really".
   I found the theorem very interesting, but I am not sure that I 
understand it correctly. I am thinking on why the theorem would not be valid
if we considered a not ordered field of complex numbers to understand the 
statement better.

  My feeling is that the theorem can be a direct consequence or a weak form
of the Frobenius theorem, but when interpreted in terms of removing 
quantifiers it got a great significance. I wonder why Sonnet said this
does not have applications. We all remember how painful it is to play with 
logical tables having quantifiers.

Cheers
Aiviet
   


On Wed, 2 Apr 1997, Ha Van Vu wrote:

> Tarsky ??
> 
> Hi,
> 
> What do you mean "closed" in the real sense ?? Best, Van
>