A generally intelligent system should be universal in the sense that it can produce arbitrary knowledge. The proof essentially shows that an algorithm cannot be generally intelligent in this sense. The reason why, after about 60 years of research, general intelligence did not show up among the systems built by the artificial intelligence community is […]

In the article SKETCH OF A PROOF: COMPUTABLE TOTAL FUNCTIONS ARE NOT ENUMERABLE I have used the notion of “total functions”. What is so special about “total functions”? First, let’s recall what a total function is. A total function is a function that assigns a value to every possible element in its domain (i.e. the set […]

Biological evolution is an example of an informal process of development, showing that such processes are actually possible in the physical world. There is certainly no single formal theory that describes how all the different genes and organisms it generated came into being. In an organism, there are some processes that are under the control […]

A formal theory cannot describe its own evolution into a more advances formal theory. Anything that is derivable inside it is so right from the start and that does not change. Some fact or formula or expression or piece of data either can be derived inside a given formal theory or it cannot. In this […]

It is a peculiar feature of the current English language that the word “formalizable” is nearly non-existent. The word feels awkward. When looking up the equivalent of the German word “formalisierbar” in one of the online-dictionaries I am using, it turned out there was no corresponding English word in that dictionary. However, there was a […]

In a previous article, I have sketched a proof that shows that the set of computable total functions (as well as the sets of programs calculating them, and of the Gödel numbers of such programs) is not enumerable. I want to look into this matter in more detail here. All of these sets are not enumerable, […]

Occasionally in this series of articles, and in this blog, I have been writing about “formal theories”. I don’t intend to introduce a formal definition of what a formal theory is here. I only want to provide an intuitive understanding. So this article is not meant for mathematicians or logicians or computer scientists, who might […]