Kolmogorov complexity of enumerating finite sets

Solovay [R.M. Solovay, On random R.E. sets, in: A.I. Arruda, N.C.A. da Costa, R. Chaqui (Eds.), Non-Classical Logics, Model Theory and Computability, North-Holland, Amsterdam, 1977, pp. 283–307] has proved that the minimal length of a program enumerating a set A is upper bounded by 3 times the negative logarithm of the probability that a random program will enumerate A. It is unknown whether one can replace the constant 3 by a smaller constant. In this paper, we show that the constant 3 can be replaced by the constant 2 for finite sets A.