b. Sandsvaer, Norway, 1887; d. Oslo, 1963 Mathematical logic
Thoralf Skolem was one of the major logicians of the twentieth century, often a lone voice in his understanding of the subtle relationship between abstract set theory and logic. He also worked on Diophantine equations and on group theory, but his contributions to mathematical logic have proved the most lasting. He taught at Bergen and Oslo, was for a time President of the Norwegian Mathematical Society and an editor of its journal, and in 1954 was named a Knight of the First Class in the Royal Order of St. Olav by the king of Norway.
In 1915 Skolem extended a result obtained by the Polish mathematician Leopold Löwenheim. His conclusion (published in 1920 and known as the Löwenheim–Skolem theorem) says that if a mathematical theory defined using only the first-order predicate calculus has a MODEL [IV.23 §1], then it has a countable model. Here a model is a set of mathematical objects that obeys the axioms of the theory. Now, the real numbers are definable in such a theory (for example, ZERMELO–FRAENKEL SET THEORY [IV.22 §3], or any other axioms for set theory). From this we obtain the so-called Skolem paradox, that the real numbers can be defined in a theory with a countable model, even though it had been known since the time of CANTOR [VI.54] that the real numbers are uncountable. How can this paradox be resolved?
The answer is that one has to be very careful about what we mean by “countable.” In this strange countable model of set theory, we can see that the reals are countable, but to the model the reals may be uncountable. In other words, the actual enumeration of the reals that we can see (i.e., the actual bijection between the reals and the natural numbers) may not belong to the model: the model can be so “small” that it is missing some functions. Skolem’s paradox highlights the difference between the viewpoint from outside the model and that from inside the model.
Several fundamental aspects of Skolem’s work are visible in these two results, the Löwenheim–Skolem theorem and the Skolem paradox. Skolem had realized, long before anyone else, that mathematical theories nearly always have several different models. He argued that there are axiom systems, and one can prove theorems in these settings, but what is meant by the objects that obey these rules will generally vary from case to case. From this he drew the radical conclusion that attempts to build mathematics on axiomatic theories were unlikely to succeed (although nowadays, of course, mathematics built on axiomatic foundations has become overwhelmingly successful).
Skolem’s insistence on first-order theories, in which variables may range only over elements, not subsets, was one that his contemporaries took time to accept. But that viewpoint, and the great clarity that comes with it, is today the overwhelmingly dominant one. Skolem insisted that the only possible logic to use in any investigation of the foundations of mathematics was FIRST-ORDER LOGIC [IV.22 §3.2], and that second-order theories were impermissible in the foundations, precisely because second-order theories allowed the axioms to refer to sets, but the nature of sets was, in his view, one of the topics to be elucidated. Skolem also felt that, while one can talk of individual objects, talk of all objects of a certain kind can be problematic if it is too informal. Indeed, a generation earlier mathematicians had encountered the paradoxes of naive set theory, where loose talk about all sets of certain kinds causes real difficulties: for example, Russell’s paradox of the set of all sets that are not members of themselves (if it is a member of itself, then it is not, but if it is not, then it is).
Skolem’s work is also characterized by a distrust of the concept of infinity and a preference for finitistic reasoning. He was an early advocate of PRIMITIVE RECURSION [II.4 §3.2.1], which deals with the theory of what are called computable functions, as a way of avoiding paradoxes concerning the infinite.
Fenstadt, J. E., ed. 1970. Thoralf Skolem: Selected Works in Logic. Oslo: Universitetsforlaget.
Jeremy Gray