Renewal point processes show up in many different fields of science and engineering. In some cases the renewal points become the only observable parts of an anticipated hidden random variation of some physical quantity. The hypothesis might be that a hidden random process originating from zero or some other low value only becomes visible at the time of first crossing of some given value level, ...

A 1976 theorem of Chaitin can be used to show that arbitrarily dense sets of lengths n have a paucity of trivial strings (only a bounded number of strings of length n having trivially low plain Kolmogorov complexities). We use the probabilistic method to give a new proof of this fact. This proof is much simpler than previously published proofs, and it gives a tighter paucity bound.

We shall prove the second incompleteness theorem via Kolmogorov complexity.

This is a short introduction to Kolmogorov complexity and information theory. The interested reader is referred to the literature, especially the textbooks [CT91] and [LV97] which cover the fields of information theory and Kolmogorov complexity in depth and with all the necessary rigor. They are well to read and require only a minimum of prior knowledge. Kolmogorov complexity. Also known as alg...