Skolem Thoralf (1887–1963), Norwegian mathematician. A pioneer of mathematical logic, he made fundamental contributions to recursion theory, set theory (in particular, the proposal and formulation in 1922 of the axiom of replacement), and model theory. His most important results for the philosophy of mathematics are the (Downward) Löwenheim-Skolem theorem (1919, 1922), whose first proof involved putting formulas into Skolem normal form; and a demonstration (1933–34) of the existence of models of (first-order) arithmetic not isomorphic to the standard model. Both results exhibit the extreme non-categoricity that can occur with formulations of mathematical theories in firstorder logic, and caused Skolem to be skeptical about the use of formal systems, particularly for set theory, as a foundation for mathematics. The existence of non-standard models is actually a consequence of the completeness and first incompleteness theorems (Gödel, 1930, 1931), for these together show that there must be sentences of arithmetic (if consistent) that are true in the standard model, but false in some other, nonisomorphic model. However, Skolem’s result describes a general technique for constructing such models. Skolem’s theorem is now more easily proved using the compactness theorem, an easy consequence of the completeness theorem. The Löwenheim-Skolem theorem produces a similar problem of characterization, the Skolem paradox, pointed out by Skolem in 1922. Roughly, this says that if first-order set theory has a model, it must also have a countable model whose continuum is a countable set, and thus apparently non-standard. This does not contradict Cantor’s theorem, which merely demands that the countable model contain as an element no function that maps its natural numbers one-toone onto its continuum, although there must be such a function outside the model. Although usually seen as limiting first-order logic, this result is extremely fruitful technically, providing one basis of the proof of the independence of the continuum hypothesis from the usual axioms of set theory given by Gödel in 1938 and Cohen in 1963. This connection between independence results and the existence of countable models was partially foreseen by Skolem in 1922.
See also CANTOR, COMPACTNESS THEOREM, GöDEL ‘s INCOMPLETENESS THEOREMS , LöW — ENHEIM -SKOLEM THEOREM , MODEL THEOR. M.H.