Related Concepts
Various model theoretic ideas are related to quantifier elimination, and there are various equivalent conditions.
Every theory with quantifier elimination is model complete.
A first-order theory T has quantifier elimination if and only if for any two models B and C of T and for any common substructure A of B and C, B and C are elementarily equivalent in the language of T augmented with constants from A. In fact, it is sufficient here to show that any sentence with only existential quantifiers have the same truth value in B and C.
Read more about this topic: Quantifier Elimination
Famous quotes containing the words related and/or concepts:
“The question of place and climate is most closely related to the question of nutrition. Nobody is free to live everywhere; and whoever has to solve great problems that challenge all his strength actually has a very restricted choice in this matter. The influence of climate on our metabolism, its retardation, its acceleration, goes so far that a mistaken choice of place and climate can not only estrange a man from his task but can actually keep it from him: he never gets to see it.”
—Friedrich Nietzsche (1844–1900)
“It is impossible to dissociate language from science or science from language, because every natural science always involves three things: the sequence of phenomena on which the science is based; the abstract concepts which call these phenomena to mind; and the words in which the concepts are expressed. To call forth a concept, a word is needed; to portray a phenomenon, a concept is needed. All three mirror one and the same reality.”
—Antoine Lavoisier (1743–1794)