From: janis_papanagnou+ng@hotmail.com   
      
   On 2026-01-27 17:31, Waldek Hebisch wrote:   
   >   
   > Actually, part that is needed here is ancient, due to Eudoksos.  Namely,   
   > real numebers a and b are equal if and only if comparing them with   
   > rational numbers gives the same result.  In other words, a is different   
   > from b if and only if there is a rational number c between a and b,   
   > but not equal to either a or b.  When computing something this   
   > principle is clumsy, but for proofs it works quit well.  Thanks   
   > to this ancients we able compute (and prove) a bunch of transcendental   
   > equalities.  Theory was finished by Dededking and Cantor who proved   
   > that number produced by limiting process exist (earlier this was   
   > consider true "by faith" or by invoking geometric intuition).   
      
   I think this needs some sorting. Speaking about "real numbers" in   
   context of Eudoksos gives a wrong impression. The ancient Greeks   
   expressed their mathematics in geometric properties and relations   
   of such entities. (Algebra, Real Numbers, etc., came much later.)   
      
   If you read modern articles you may find references to real numbers   
   in context of ancient Greek mathematics, but these are only used to   
   explain that old knowledge with modern methods.   
      
   Janis   
      
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