Skolem published around 180 papers on
Diophantine equations,
group theory,
lattice theory, and,
set theory and
mathematical logic, with some 50 papers in the last two topics. 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. Skolem in these works used the terminology of
Ernst Schröder's algebraic logic with the effect that his results were not understood. By 1919, he had defined
Heyting algebras under the name
Gruppenkalkül and established its basic properties, thus anticipating intuitionistic propositional logic. After some of his results were rediscovered by others, Skolem published a 1936 paper "" in German, surveying his earlier work in lattice theory, but this paper was not well received by those whose work he had anticipated. It was found out in the early 1990's that Skolem's paper of 1920, with the enormously long title
Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze, nebst einem Theorem über dichte Mengen, contained a polynomial-time decision method for the
word problem for freely generated lattices, a result otherwise known only since 1988. Skolem was a pioneer
model theorist. In 1920, in the mentioned long-titled article, 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 countable 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
Kőnig'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 Π, Σ as variable-binding quantifiers, in contrast to the notations of
Peano,
Principia Mathematica, and
Principles of Mathematical Logic. In 1919, Skolem read the
Principia Mathematica but was not satisfied with one aspect of Russell's treatment, namely the universal and existential quantification over infinite domains. His answer was to develop
primitive recursive arithmetic. His paper, again with a long title, is:
Begründung der elementary Arithmetic durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlichen mit unendlichem Ausdehnungsbereich. (A foundation for elementary arithmetic through the recurrent mode of thought, without the use of bound variables over an infinite domain.) This paper got published only in 1923, after having been rejected by the premier Stockholm-based journal
Acta Mathematica, an event that embittered Skolem greatly, as did the appropriation of his discovery by
David Hilbert. Failed acknowledgement of his work in the Scandinavian countries led him to work for many years mainly on Diophantine problems. In 1929, Skolem anticipated Gödel's
incompleteness theorem of 1931, writing: It would be an interesting task to show that every collection of propositions about the natural numbers, formulated in first- order logic, continues to hold when one makes certain changes in the meaning of “numbers.” After Gödel's publication, Skolem (1934) carried through the result anticipated in 1929, by 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. == Completeness ==