unbounded quantifier造句

• The first level of the Levy hierarchy is defined as containing only formulas with no unbounded quantifiers, and is denoted by \ Delta _ 0 = \ Sigma _ 0 = \ Pi _ 0.
• Thus there is a formula \ varphi ( n, n _ 1 ) in first-order arithmetic with no unbounded quantifiers, such that T halts on input n at time n 1 at most if and only if \ varphi ( n, n _ 1 ) is satisfied.
• Prefix induction can simulate predecessor induction, but only at the cost of making the statement more syntactically complex ( adding a bounded universal quantifier ), so the interesting results relating prefix induction to polynomial-time computation depend on excluding unbounded quantifiers entirely, and limiting the alternation of bounded universal and existential quantifiers allowed in the statement.
• The consequent most immediate challenge in Geometry is to  correct Euclid by establishing the theorems of Book VI on the basis of the methods and techniques of Books I-IV, avoiding the use of the ratio concept introduced in Book V . For Arithmetic the corresponding challenge is to establish the results of Book VII-IX without resort to the sort of iterative procedure that Euclid allows himself in the definition of multiplication . ( Book VII, Definition 15 . ) For Cantorian Arithmetic the main challenge would be to show that the great body of infinitary mathematics-the disciplines flowing in one way or another from the calculus-does not require unbounded quantifiers and consequently that the instances of the Replacement Schema of the Zermelo-Fraenkel axioms for set theory involving such quantifiers, are, as well as being disallowed by Mayberry s general philosophy, in any case technically redundant.