From complexity to incompleteness of treatable information

The necessary information for specifying a complex system may not be completely accessible to us, i.e., to mathematical treatments. This is not to be confounded with the incompleteness of our knowledge about whatever systems or nature, since here information is our ignorance. In conventional statistics and information theories, this information or ignorance is supposed completely accessible to theoretical treatments connected with complete probability distributions. However, the hypothesis of incomplete information supposes that the information of certain systems can be incomplete as calculated in the usual way as in the conventional information theories. This hypothesis has been used in order to generalize the conventional statistics theory. The generalized statistics and information theory characterized by an empirical parameter has been proved useful for the formulation of the nonextensive statistical mechanics based on Tsallis entropy, for the description of some correlated quantum systems and for the derivation of the stationary probability distributions of nonequilibrium complex systems evolving in hierarchical or fractal phase space. In this paper, the incompleteness of the information will be discussed from mathematical, physical and epistemological considerations with an example of information calculation in fractal phase space with stationary probability distribution.