  Although G\"odel's incompleteness theorem made mathematician recognize that
no axiomatic system could completely prove its correctness and that there is an
eternal hole between our knowledge and the world, physicists so far continue to
work on the approaches based on the hypothesis to completely or approximately
know the systems of interest. In this paper, however, I review the recent
development of a different approach, a statistical theory based upon the notion
of incomplete information. This consideration leads to generalized statistical
mechanics characterized by an incompleteness parameter which equals unity when
information is complete. The mathematical and physical bases of the information
incompleteness are discussed. The application of the concomitant incomplete
quantum distribution to correlated electron systems is reviewed.
