PlanetMath: quantifier free

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quantifier free

(Definition)

Let be a first order language. A formula $\psi$ is quantifier free iff it contains no quantifiers.

Let be a complete -theory. Let $S \subseteq L$ . Then is an elimination set for iff for every $\psi(\bar{x}) \in L$ there is some $\phi(\bar{x}) \in S$ so that $T \vdash \forall \bar{x} (\psi(\bar{x})) \leftrightarrow \phi(\bar{x})$ .

In particular, has quantifier elimination iff the set of quantifier free formulas is an elimination set for . In other words has quantifier elimination iff for every $\psi(\bar{x}) \in L$ there is some quantifier free $\phi(\bar{x}) \in L$ so that $T \vdash \forall \bar{x} (\psi(\bar{x})) \leftrightarrow \phi(\bar{x})$ .

"quantifier free" is owned by mathcam. [ full author list (2) | owner history (1) ]

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See Also: quantifier, logical language

Also defines:

quantifier free formula, quantifier elimination, elimination set

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Cross-references: complete, quantifiers, contains, iff, formula, first order language
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This is version 4 of quantifier free, born on 2003-02-12, modified 2007-01-11.
Object id is 4031, canonical name is QuantifierFree.
Accessed 5535 times total.

Classification:

AMS MSC:	03B10 (Mathematical logic and foundations :: General logic :: Classical first-order logic)
	03C07 (Mathematical logic and foundations :: Model theory :: Basic properties of first-order languages and structures)
	03C10 (Mathematical logic and foundations :: Model theory :: Quantifier elimination, model completeness and related topics)

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