FOM: Quantifier question

[Vladik Kreinovich]

Just FYI: It is known that if we add a quantifier "there exists uncountable 
many" to elementary geomtery (in the sense of the famous Tarski-Seidenberg 
result of decidability of the first order theory of real numbers) then the 
theory remians decidable:

   Rapp, Andreas. {\it The ordered field of real numbers and logics with
Malitz quantifiers}, The Journal of Symbolic Logic, 1985, Vol. 50,
No. 2, pp. 380--389.

We used this result in our paper:

 Vladik Kreinovich and Olga Kosheleva, "An application of
  logic
  to combinatorial geometry: how many tetrahedra are equidecomposable
  with a cube?".  Mathematical Logic Quarterly, 1994, Vol. 40, No.
  1, pp. 31-34.