Quantifier Elimination and Real Closed Ordered Fields with a Predicate for the Powers of Two

In this thesis we first review the model theory of quantifier elimination and investigate the logical relations among various quantifier elimination tests. In particular we prove the equivalence of two quantifier elimination tests for countable theories. Next we give a procedure for eliminating quantifiers for the theory of real closed ordered fields with a predicate for the powers of two. This result was first obtained by van den Dries [20]. His method is model-theoretic, which provides no apparent bounds on the complexity of a decision procedure. In the last section we give a complete axiomatization of the theory of real closed ordered fields with a predicate for the Fibonacci numbers.