[FOM] Primitive symbols in the theory of real closed fields

[Rafael Grimson]

Adding the binary relation symbol "<" to the vocabulary <+, x, 0, 1>
Tarski [Tar51] obtained a theory, R, for real closed fields that
admits quantifier elimination (and the symbol < is insdispensable to
obtain this).

In his article [Tar31] Tarksi also shows that the product symbol, x,
is indispensable, i.e., the set of formulas that do not involve this
symbol has less expressivle power than the complete language.

Does someone know if it is proven that the addition symbol is also
indispensable?

Best egards,
Rafael Grimson



[Tar31] A. Tarski, Sur les ensembles denissables de nombres reels, Fundamenta
Mathematicae 17 (1931), 210-239.
[Tar51] , A decision method for elementary algebra and geometry, second
ed., Univ. of California Press, Berkley, CA, 1951.