In this project we examine the question of whether any algorithmical resp. non-algorithmical concept as well as any kind of scientific theory are ever be incomplete.  We investigate this question with particular attention to the claims made that these concepts can contribute to the notions of observation, prediction, recollection and explanation.  There’s some evidence about an interrelation between ideas within the philosophy of science – the Duhem-Quine thesis of underdetermination of observations and the observational/theoretical terms distinction failure – and the well known limitative theorems of Gödel and Tarski etc.  Although these results original apply to deduction from axioms, we have a further evidence to assume that their extensions hold for general inference devices, i.e. recursive enumerable ones (like structural inductive) as well, as for other possible systems of logic with non-recursive sets of axioms resp. rules of inference and furthermore for any constraint satisfaction problem. This would imply that any idea or concept and any experience cannot be completely defined or contextualized. – So there’s a strong sense in which we remain under the shadow of chance and randomness.

The aim of this research work is to review, clarify, and critically analyze aspects of modern mathematical information theories.  The emphasis is upon mathematical structures involved, rather than numerical computations.  We will argue that theories and concepts of information and complexity can never be complete.  For that reason particular attention will be paid to various provided measures of information and complexity and their dependence on algorithmical, resp. non-algorithmical concepts.  We try to reveal the supposed conditions for incompleteness like computational irreducibility, arbitrariness, infinity, and self-awareness.  Working hypothesis is that due to connections with disguised forms of the meta-mathematical theorems of Gödel and Tarski incompleteness is widely an epistemological limit which is manifest, e.g. in the non-existence of a procedure to determine valid empirical observations resp. the undefinability of valid observations, and we assume that this limit is not likely to be broken any time soon.