The Berry-like Sentence in the First-order Peano Arithmetic System with the Operation of Fa.pdf

1 The Berry-like Sentence in the First-order Peano Arithmetic System with the Operation of Factorial T. Mei (Central China Normal University, Wuhan, Hubei PRO, People’s Republic of China E-Mail: meitao@ meitaowh@ ) Abstract: A first-order Peano Arithmetical system with the operation of factorial (PAF) is introduced. For any formula A(x) with a free variable x in PAF, we define a corresponding B-formula ? xBA(x) so as to there exists unique number that is smallest in all natural numbers satisfying the formula A(x) that satisfies the formula BA(x) if A(x) is satisfiable. And then, we construct a formula ? xBD(x) which means that “there exists x, for any B-formula ? xB’A(x) whose G?del code is smaller than a constant a, x does not satisfy B’A(x), and x is the smallest in those numbers that have such character.” However, ? xBD(x) itself is a B-formula and its G?del code is smaller than a. Thus, ? xBD(x) is a version in PAF of the Berry sentence “The smallest positive integer not nameable in under eleven words” that itself is in only ten words . 1 Basic idea A version of the Berry paradox[1, 2] arises from considering a sentence “The smallest positive integer not nameable in under eleven words.” But the sentence itself is a specification for that number in only ten words. The application of the Berry paradox in the information-theoretic approach had been discussed by G. J. Chaitin[3]. In this paper, we try to construct a Berry-like sentence in the first-order Peano Arithmetic system with the operation of factorial (PAF). The essential steps are as follows: (1) For any formula A(x) with a free variable x, we construct a corresponding formula BA(x) with a free variable x: BA(x): A(x) ∧ ? y A(y)→((y≠ x)→(yx)) The above formula has many equivalent forms. However, we prescribe that the above standard form is only considered in spite of another equivalent forms. Any formula with a free variable x is false or satisfiable. And mayb