Thoralf Skolem - Completeness

Completeness

The completeness of first-order logic is an easy corollary of results Skolem proved in the early 1920s and discussed in Skolem (1928), but he failed to note this fact, perhaps because mathematicians and logicians did not become fully aware of completeness as a fundamental metamathematical problem until the 1928 first edition of Hilbert and Ackermann's Principles of Mathematical Logic clearly articulated it. In any event, Kurt Gödel first proved this completeness in 1930.

Skolem distrusted the completed infinite and was one of the founders of finitism in mathematics. Skolem (1923) sets out his primitive recursive arithmetic, a very early contribution to the theory of computable functions, as a means of avoiding the so-called paradoxes of the infinite. Here he developed the arithmetic of the natural numbers by first defining objects by primitive recursion, then devising another system to prove properties of the objects defined by the first system. These two systems enabled him to define prime numbers and to set out a considerable amount of number theory. If the first of these systems can be considered as a programming language for defining objects, and the second as a programming logic for proving properties about the objects, Skolem can be seen as an unwitting pioneer of theoretical computer science.

In 1929, Presburger proved that Peano arithmetic without multiplication was consistent, complete, and decidable. The following year, Skolem proved that the same was true of Peano arithmetic without addition, a system named Skolem arithmetic in his honor. Gödel's famous 1931 result is that Peano arithmetic itself (with both addition and multiplication) is incompletable and hence a posteriori undecidable.

Hao Wang praised Skolem's work as follows:

"Skolem tends to treat general problems by concrete examples. He often seemed to present proofs in the same order as he came to discover them. This results in a fresh informality as well as a certain inconclusiveness. Many of his papers strike one as progress reports. Yet his ideas are often pregnant and potentially capable of wide application. He was very much a 'free spirit': he did not belong to any school, he did not found a school of his own, he did not usually make heavy use of known results... he was very much an innovator and most of his papers can be read and understood by those without much specialized knowledge. It seems quite likely that if he were young today, logic... would not have appealed to him." (Skolem 1970: 17-18)

For more on Skolem's accomplishments, see Hao Wang (1970).