Definitions

Bounded quantifier

In the study of formal theories in mathematical logic, bounded quantifiers are often added to a language. These are two quantifiers in addition to $forall$ and $exists$. They are motivated by the fact that determining whether a sentence with only bounded quantifiers is true is not as difficult as determining whether an arbitrary sentence is true.

Bounded quantifiers in arithmetic

Suppose that L is the language of Peano arithmetic (the language of second-order arithmetic or arithmetic in all finite types would work as well). There are two bounded quantifiers: $forall n < t$ and $exists n < t$. These quantifiers bind the number variable n and contain a numeric term t which may not mention n but which may have other free variables.

The semantics of these quantifiers is determined by the following rules:

$exists n < t, phi Leftrightarrow exists n \left(n < t land phi\right)$
$forall n < t, phi Leftrightarrow forall n \left(n < t rightarrow phi\right)$

There are several motivations for these quantifiers.

• In applications of the language to recursion theory, such as the arithmetical hierarchy, bounded quantifiers add no complexity. If $phi$ is a decidable predicate then $exists n < t , phi$ and $forall n < t, phi$ are decidable as well.
• In applications to the study of Peano Arithmetic, formulas are sometimes provable with bounded quantifiers but unprovable with unbounded quantifiers.

For example, there is a definition of primality using only bounded quantifers. A number n is prime if and only if there are not two numbers strictly less than n whose product is n. There is no quantifier free definition of primality in the language $langle 0,1,+,times, <, =rangle$, however. The fact that there is a bounded quantifier formula defining primality shows that the primality of each number can be computably decided.

In general, a relation on natural numbers is definable by a bounded formula if and only if it is computable in the linear-time hierarchy, which is defined similarly to the polynomial hierarchy, but with linear time bounds instead of polynomial. Consequently, all predicates definable by a bounded formula are Kalmár elementary, context-sensitive, and primitive recursive.

In the arithmetical hierarchy, an arithmetical formula which contains only bounded quantifiers is called $Sigma^0_0$, $Delta^0_0$, and $Pi^0_0$. The superscript 0 is sometimes omitted.

Bounded quantifiers in set theory

Suppose that L is the language $langle in, ldots, =rangle$ of set theory, where the ellipsis may be replaced by term-forming operations such as a symbol for the powerset operation. There are two bounded quantifiers: $forall x in t$ and $exists x in t$. These quantifiers bind the set variable x and contain a term t which may not mention x but which may have other free variables.

The semantics of these quantifiers is determined by the following rules:

$exists x in t, phi Leftrightarrow exists x \left(x in t land phi\right)$
$forall x in t, phi Leftrightarrow forall x \left(x in t rightarrow phi\right)$

A formula of set theory which contains only bounded quantifiers is called Δ0.

Bounded quantifiers are important in Kripke-Platek set theory and constructive set theory, where only Δ0 separation is included. That is, it includes separation for formulas with only bounded quantifiers, but not separation for other formulas. In KP the motivation is the fact that whether a set x satisfies a bounded quantifier formula only depends on the collection of sets that are close in rank to x (as the powerset operation can only be applied finitely many times to form a term). In constructive set theory, it is motivated on predicative grounds.

References

• Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-568-81262-0.
• Kunen, K. (1980). Set theory: An introduction to independence proofs. Elsevier. ISBN 0-444-86839-9.

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