  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.
