History
In early model theory, quantifier elimination was used to demonstrate that various theories possess certain model-theoretic properties like decidability and completeness. A common technique was to show first that a theory admits elimination of quantifiers and thereafter prove decidability or completeness by considering only the quantifier-free formulas. This technique is used to show that Presburger arithmetic, i.e. the theory of the additive natural numbers, is decidable.
Theories could be decidable yet not admit quantifier elimination. Strictly speaking, the theory of the additive natural numbers did not admit quantifier elimination, but it was an expansion of the additive natural numbers that was shown to be decidable. Whenever a theory in a countable language is decidable, it is possible to extend its language with countably many relations to ensure that it admits quantifier elimination (for example, one can introduce a relation symbol for each formula).
Example: Nullstellensatz in ACF and DCF.
Read more about this topic: Quantifier Elimination
Famous quotes containing the word history:
“Literary works cannot be taken over like factories, or literary forms of expression like industrial methods. Realist writing, of which history offers many widely varying examples, is likewise conditioned by the question of how, when and for what class it is made use of.”
—Bertolt Brecht (1898–1956)
“Racism is an ism to which everyone in the world today is exposed; for or against, we must take sides. And the history of the future will differ according to the decision which we make.”
—Ruth Benedict (1887–1948)
“Modern Western thought will pass into history and be incorporated in it, will have its influence and its place, just as our body will pass into the composition of grass, of sheep, of cutlets, and of men. We do not like that kind of immortality, but what is to be done about it?”
—Alexander Herzen (1812–1870)