Thoralf Skolem - Mathematics

Mathematics

Skolem published around 180 papers on Diophantine equations, group theory, lattice theory, and most of all, set theory and mathematical logic. He mostly published in Norwegian journals with limited international circulation, so that his results were occasionally rediscovered by others. An example is the Skolem–Noether theorem, characterizing the automorphisms of simple algebras. Skolem published a proof in 1927, but Emmy Noether independently rediscovered it a few years later.

Skolem was among the first to write on lattices. In 1912, he was the first to describe a free distributive lattice generated by n elements. In 1919, he showed that every implicative lattice (now also called a Skolem lattice) is distributive and, as a partial converse, that every finite distributive lattice is implicative. After these results were rediscovered by others, Skolem published a 1936 paper in German, "Über gewisse 'Verbände' oder 'Lattices'", surveying his earlier work in lattice theory.

Skolem was a pioneer model theorist. In 1920, he greatly simplified the proof of a theorem Leopold Löwenheim first proved in 1915, resulting in the Löwenheim–Skolem theorem, which states that if a first-order theory has an infinite model, then it has a countable model. His 1920 proof employed the axiom of choice, but he later (1922 and 1928) gave proofs using König's lemma in place of that axiom. It is notable that Skolem, like Löwenheim, wrote on mathematical logic and set theory employing the notation of his fellow pioneering model theorists Charles Sanders Peirce and Ernst Schröder, including ∏, ∑ as variable-binding quantifiers, in contrast to the notations of Peano, Principia Mathematica, and Principles of Mathematical Logic. Skolem (1934) pioneered the construction of non-standard models of arithmetic and set theory.

Skolem (1922) refined Zermelo's axioms for set theory by replacing Zermelo's vague notion of a "definite" property with any property that can be coded in first-order logic. The resulting axiom is now part of the standard axioms of set theory. Skolem also pointed out that a consequence of the Löwenheim–Skolem theorem is what is now known as Skolem's paradox: If Zermelo's axioms are consistent, then they must be satisfiable within a countable domain, even though they prove the existence of uncountable sets.