Note that any o-minimal structure has quantifier elimination if we also allow free use of parameters. The classic example is Rexp = (R; <, +, −, ×, 0, 1,ex). Wilkie showed that the …

To show constructively that a theory has quantifier elimination, it suffices to show that we can eliminate an existential quantifier applied to a conjunction of literals, that is, show that each …

To prove quantifier elimination, it suffices to show that if is a conjunction of atomic and negated-atomic formulas, and is a single variable, then is equivalent to a quantifier-free formula.

Since quantifier elimination is equivalent to substructure completeness and elementarily equivalent finite structures are isomorphic, (the theory of a) finite first-order structure …

Quantifier Elimination Theorem The theory of dense linear orders admits elimination of quantifiers. Proof Consider a formula ' = 9 x ( ^ ^ 0 l), where each i is onl

Quantifier Elimination 2.1 Elimination sets Let L be a language. It may happen that two different L-formulas <p( iT) and <p' (v) admit the same meaning in a structure A of L, or in a class of L …

If the theory Σ(N,<) Σ (N, <) were to admit elimination of quantifiers then every formula, including ϕ ϕ, should be preserved under embeddings. But if we consider the embedding f: x ↦ x + 1 f: x …

1 Quantifier elimination A quantifier free formula is an expression consisting of polynomial equations (f (x) = 0) and inequalities (or), and ⇒ (implies). We often also allow strict …

In short, the theory of this model Th ( Q, < ) admits elimination of quantifiers. What about other endless dense linear orders? The argument we have given so far is about the theory of this …

There is a quanti er-free L-formula (v) such that T j= 8v ( (v) $ (v)). If M and N are models of T, A is an L-structure, A M, and A N, then M j= (a) if and only if N j= (a) for all a 2 A.