Hypothesis testing and the error of the third kind

In this note it is shown that the concept of an error of the third kind (type-III-error) stems from a wrong understanding of the concept of hypothesis testing in mathematical statistics. If the alterna-tive hypothesis in a statistical testing problem states that the null hypothesis is wrong, then an error of the third kind cannot occur. This error can only be defined in more-decision-problems, i.e. prob-lems where we have to decide for one of more than two possibilities. Further the problem of the power of tests between a null and an alternative hypothesis is discussed. The notation of set theory is used because it allows for a concise notation of this type of statistical testing problems. The notation is explained in an appendix.