Bounded quantifier

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 types of bounded quantifiers: ${\displaystyle \mathrm{\forall }n<t}$ and ${\displaystyle \mathrm{\exists }n<t}$. These quantifiers bind the number variable n using a numeric term t not containing n but which may have other free variables. ("Numeric terms" here means terms such as "1 + 1", "2", "2 × 3", "m + 3", etc.)

These quantifiers are defined by the following rules (${\displaystyle \varphi }$ denotes formulas):

${\displaystyle \mathrm{\exists }n<t\varphi \iff \mathrm{\exists }n(n<t\wedge \varphi )}$

${\displaystyle \mathrm{\forall }n<t\varphi \iff \mathrm{\forall }n(n<t\to \varphi )}$

There are several motivations for these quantifiers.

• In applications of the language to computability theory, such as the arithmetical hierarchy, bounded quantifiers add no complexity. If ${\displaystyle \varphi }$ is a decidable predicate then ${\displaystyle \mathrm{\exists }n<t\varphi }$ and ${\displaystyle \mathrm{\forall }n<t\varphi }$ are decidable as well.
• In applications to the study of Peano arithmetic, the fact that a particular set can be defined with only bounded quantifiers can have consequences for the computability of the set. For example, there is a definition of primality using only bounded quantifiers: 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 ${\displaystyle ⟨0,1,+,\times ,<,=⟩}$, 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 that contains only bounded quantifiers is called ${\displaystyle {\mathrm{\Sigma }}_0^0}$, ${\displaystyle {\mathrm{\Delta }}_0^0}$, and ${\displaystyle {\mathrm{\Pi }}_0^0}$. The superscript 0 is sometimes omitted.

Bounded quantifiers in set theory

Suppose that L is the language ${\displaystyle ⟨\in ,\dots ,=⟩}$ of the Zermelo–Fraenkel 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: ${\displaystyle \mathrm{\forall }x\in t}$ and ${\displaystyle \mathrm{\exists }x\in t}$. 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:

${\displaystyle \mathrm{\exists }x\in t(\varphi )\iff \mathrm{\exists }x(x\in t\wedge \varphi )}$

${\displaystyle \mathrm{\forall }x\in t(\varphi )\iff \mathrm{\forall }x(x\in t\to \varphi )}$

A ZF formula that contains only bounded quantifiers is called ${\displaystyle {\mathrm{\Sigma }}_0}$, ${\displaystyle {\mathrm{\Delta }}_0}$, and ${\displaystyle {\mathrm{\Pi }}_0}$. This forms the basis of the Lévy hierarchy, which is defined analogously with the arithmetical hierarchy.

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.