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:
“One classic American landscape haunts all of American literature. It is a picture of Eden, perceived at the instant of history when corruption has just begun to set in. The serpent has shown his scaly head in the undergrowth. The apple gleams on the tree. The old drama of the Fall is ready to start all over again.”
—Jonathan Raban (b. 1942)
“The basic idea which runs right through modern history and modern liberalism is that the public has got to be marginalized. The general public are viewed as no more than ignorant and meddlesome outsiders, a bewildered herd.”
—Noam Chomsky (b. 1928)
“Regarding History as the slaughter-bench at which the happiness of peoples, the wisdom of States, and the virtue of individuals have been victimized—the question involuntarily arises—to what principle, to what final aim these enormous sacrifices have been offered.”
—Georg Wilhelm Friedrich Hegel (1770–1831)