The Berry-like Sentence in the First-order Peano Arithmetic System with the Operation of Factorial

Details The Berry-like Sentence in the First-order Peano Arithmetic System with the Operation of Factorial The Berry-like Sentence in the First-order Peano Arithmetic System with the Operation of Factorial

2006-07-19

arxiv.org/abs/math/0607459v1

Mathematics

Logic

7 pages, 0 figures

Scientific paper

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 which means that there exists unique number that is smallest in all natural numbers satisfying the formula A(x) that satisfies the B-formula if A(x) is satisfiable. And then, we construct a formula which means that "there exists x, for any B-formula whose Godel code is smaller than a constant a, x does not satisfy this B-formula, and x is the smallest in those numbers that have such character." However, the constructed formula itself is a B-formula and its Godel code is smaller than a. Thus, it 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.

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