2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/115037A 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.7 pages, 0 figuresLogictext