Bounded quantifiers

In the study of formal theories in mathematical logic, bounded quantifiers are often included in a formal language in addition to the standard quantifiers "∀" and "∃". Bounded quantifiers differ from "∀" and "∃" in that bounded quantifiers restrict the range of the quantified variable. The study of bounded quantifiers is motivated by the fact that determining whether a sentence with only bounded quantifiers is true is often not as difficult as determining whether an arbitrary sentence is true.

Examples of bounded quantifiers in the context of real analysis include "∀x>0", "∃y<0", and "∀x ∊ ℝ". Informally "∀x>0" says "for all x where x is larger than 0", "∃y<0" says "there exists a y where y is less than 0" and "∀x ∊ ℝ" says "for all x where x is a real number". For example, "∀x>0 ∃y<0 (x = y²)" says "every positive number is the square of a negative number".

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: ${\displaystyle \forall n<t}$ and ${\displaystyle \exists n<t}$. 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. ("Numeric terms" here means terms such as "1 + 1", "2", "2 × 3", "m + 3", etc.)

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