The Norwegian mathematician Thoralf Skolem (1887-1963) is not a well-known figure, despite the number and importance of his contributions. He was the author of some 180 articles on diophantine equations, group theory, lattice theory, combinatorics and, not least, mathematical logic and set theory. But he published mainly in Nordic journals with a small circulation, which explains why his work had less immediate impact than it deserved. One example is the Skolem-Noether theorem, which gives a characterisation of automorphisms of simple algebras: the famous mathematician Emmy Noether rediscovered it independently, but Skolem had already published a proof in 1927.
Skolem’s work is especially well known among logicians, his name being attached to the famous Löwenheim-Skolem theorem, the first metalogical result to be proved, and to the “Skolem paradox”. He was a pioneer in the study of non-standard models, undoubtedly one of the best logicians of the first half of the 20th century, although his critical and somewhat sceptical ideas about the foundations perhaps also detracted from his impact.
The usual axiomatic system of set theory (known by the acronym ZFC) is called Zermelo-Fraenkel, but it could well have been called Zermelo-Skolem. In a 1922 lecture, which is the subject of this note, Skolem refined the axioms proposed by Zermelo, using first-order logic to refine the idea of “definite property”. He also pointed out the need to add the axiom of replacement in order to generate all of Cantor’s alephs. Abraham Fraenkel made comparable (though less clear and less successful) contributions in several papers in the 1920s. If in the end the axiom system bears one name and not the other, it is probably because Fraenkel was a great enthusiast of set theory and wrote a handbook that did much to spread it. As for Skolem’s attitude, we shall see it in a moment.
A 1920 article entitled ‘Logico-combinatorial investigations on satisfiability…’ begins by simplifying and generalising the proof of a theorem presented by the German logician Löwenheim in 1915. Given a first-order theory (i.e., a denumerable set of propositions), if that theory has a model A, then it already has a model whose domain is denumerable. Skolem’s proof used the axiom of choice to prove that there is in fact a submodel of A that satisfies the statement of the theorem. A couple of years later, he gave a new proof of his theorem that does not employ the axiom of choice, and applied it to axiomatic set theory itself.
By refining Zermelo’s system of axioms, we obtain the ZFC system formulated in first-order logic. And thus the Löwenheim-Skolem theorem is fully applicable to it, which gives rise to a paradoxical situation: the ZFC axioms are intended to ground Cantorian set theory, which is eminently a theory of infinite sets, most of which are non-denumerable (the elementary example is R, the set of real numbers); and yet ZFC theory is already satisfied in a denumerable model. The entire universe of sets – insofar as its existence is determined by the axioms – could be nothing but a denumerable domain. The theory proves the ‘existence’ of non-denumerable sets, but such sets can live (if the expression is permitted) in a domain containing only a denumerable quantity of things.
This is not a paradox like Russell’s or Cantor’s paradoxes, which are in fact contradictions of a logicist set theory. Skolem’s paradox is a true paradox, that is, a result which strongly contradicts our expectations, but which does not give rise to a contradiction. It was Skolem himself, in his masterly article, who offered the standard explanation of why there is no contradiction.
In abandoning informal mathematics for axiomatics, ‘set’ is no longer an arbitrary collection, but an object that is linked to other objects in the domain through certain relations postulated in the axioms. It is therefore not contradictory that an object A of the domain is non-denumerable in the sense of the axioms — within the domain there is no ‘bijection’ between A and the set of naturals – and that at the same time the whole domain is denumerable — and therefore all the objects that play the role of ‘elements’ of A are denumerable. From outside the domain one can recognise a collection of pairs that establishes a bijection between the elements of A and the naturals, but that collection is simply not a ‘set’, i. e., it is not part of the domain. Of course, in the domain we will have, in addition to N, another object pN which is the ‘set’ of all its subsets, but this ‘set’ is only non-denumerable when viewed from within the domain: all its ‘elements’ are clearly denumerable because they belong to a countable model.
After posing Skolem’s paradox, the inevitable result of the complete formalisation of the axiomatic system, and explaining its nature, the Norwegian drew his main consequence: “axiomatising set theory leads to a relativity of set-theoretic notions, and this relativity is inseparably linked to full axiomatisation”.
Skolem confessed his surprise that so many mathematicians found in the ZFC axioms an “ideal foundation” for mathematics, when in his opinion this is not the case at all. Key notions such as ‘denumerable’ and ‘non-denumerable’ are relativised; even ‘finite’ and ‘infinite’ are affected. This is why Skolem declared himself in favour of a foundation that works with more natural and immediately clear notions, such as the concept of the integers and inferences by mathematical induction.
In other, also very important works, Skolem developed (without using quantifiers) primitive recursive arithmetic, which would be so fruitful in the hands of Kurt Gödel, and he was the first to demonstrate the existence of non-standard models of formalised arithmetic.
Skolem was not a dogmatist, but he was convinced that the finite treatment of mathematics is the most rigorous starting point. Probably more dogmatic is the attitude of those who attribute an absolute meaning to the idea of the ‘super-denumerable’ (and here we should recall the undecidability of Cantor’s continuum hypothesis, a result that Skolem suspected as early as 1922, as we know from a footnote to his article). However, this finitistic position is seen by many as too extreme, and its motivations as too philosophical. Finitism remained a minority position, but Skolem’s contributions are key elements of mathematical logic.
His biographer Jens Fenstad tells us that Skolem was very modest and reserved. He was a very creative mathematician, interested in new fields. He did not create a school, although he inspired more than one of the younger Norwegian mathematicians with his great achievements. After retirement, he continued to be very active as a researcher, publishing and visiting American universities on several opportunities.
Learn more:
Skolem’s key articles quoted above can be found in J. van Heijenoort’s compilation, From Frege to Gödel (1967, 2002).
Selected Works in Logic, published by Jens E. Fenstad (Oslo 1970), is long out of print, but a biographical sketch of Fenstad can be found in a digital version.
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