Time of crisis

“Adversity makes us wiser; prosperity destroys our appreciation of what is fair.” – Seneca.

“We are in a time of crisis”: a phrase that is repeated, perhaps by every human generation. Who are we? Who are you? In times of crisis, says R. Solnit, these are matters of life and death: after an earthquake, a bomb attack, or during a serious epidemic, most people are altruistic, caring for those around them and not only for themselves – whether they are friends, loved ones or strangers. And in days of crisis, it never hurts to remember that, in the end, such situations are relatively normal.

Yet I don’t want to write about the current crisis, but about something that came to my mind when I was talking to some students about events of just over a century ago. A serious time of crisis, no doubt, the years around 1920: Europe was coming out of a terrible experience that made fundamental changes to that society and that culture. And the fact is, the changes reached even to the most abstract and apparently remote spheres of intellectual activity. They reached mathematics.

I would like to mention three events of those years around 1920, and focus especially on the relatively less known works of Thoralf Skolem (1887-1963).

In 1921, H. Weyl wrote an article on “the new foundational crisis in mathematics” and diagnosed a serious “vicious circle” in the foundations of analysis. He himself had tried to outline a way out of these circularities in the 1918 book Das Kontinuum (The Continuum), renouncing the classical system of real numbers, accepting a mismatch between the intuitive continuum and the number system, and dropping the supremum principle. (Principle of the supremum: given a set C of real numbers, bounded above, there exists a lower upper bound, or supremum, of C.) The definition of the supremum turns out to be impredicative, and therefore the principle is discarded, but it is possible to reconstruct various results of analysis by way of predicativism. The rest, what cannot be reconstructed, is sacrificed for the sake of progress.

But in 1921 it seemed to Weyl that an even more radical solution, intuitionism, was even better. “Brouwer is the revolution”, he wrote, in a sentence that seems to have sent a shiver down Hilbert’s spine: a genuine fear that young mathematicians might abandon forever the path of classical mathematics, analysis, and set theory. These were times of radicalism, when everyone seemed ready to abandon the old ways in favour of the new. The world of the bourgeois was decadent and outdated, it had to be replaced by ‘something else’: communism, anarchism, fascism, whatever – but something else.

Good old Hilbert, a bourgeois after all, decided to defend the more conservative approach of “Cantor’s paradise”.

L. E. J. Brouwer (Bertus to his friends) had begun to develop an “intuitionistic set theory”, in which the principle of the excluded middle ceased to govern logic; a theory which would later be called “intuitionistic mathematics” and which claimed to ensure intuitiveness and certainty in a way that the mathematics of real numbers and transfinite cardinals certainly did not. This new theory, much more difficult to handle than the classical one (“the spheres of truth are less transparent than those of illusion”, he said in 1933), led to surprising results. Two examples: not every real number admits a development into decimals, but every total function is uniformly continuous.

While Weyl and Brouwer explored the difficult paths of predicativism and intuitionism, a lesser-known mathematician, the Norwegian Thoralf Skolem, developed the ideas of the new mathematical logic with great rigour, and showed in passing that the set-theoretic foundation of mathematics had serious shortcomings. The key work is his outstanding 1922 lecture, ‘Some Remarks on Axiomatic Set Theory’, which unfortunately was published in a somewhat obscure place and was much less widely read than it deserved to be.

In this article, Skolem expressed his surprise that people saw axiomatic set theory as the foundation of mathematics, and gave his reasons for being skeptical. “I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics…”

He began by refining Zermelo’s axiomatic theory, which at that time was not yet a fully formalized system: he refined the Axiom of Separation by relying on first-order logic, and raised the need to add the Axiom of Replacement. Essential additions to the theory. It should be said that the Zermelo-Fraenkel axioms (the so-called ZFC system) should rather be called Zermelo-Skolem axioms, considering the quality and clarity of Skolem’s contribution…

Once this is done, we have an impeccable axiomatic system, but one that is far from absolutely determining the key concepts of the theory. Even such fundamental notions as denumerable and non-denumerable turn out relative. Why? Because first-order ZFC theory is impacted by the Löwenheim-Skolem Theorem: if a theory in first-order language has an infinite model, then it already has a denumerable model. This means that ZFC is unable to determine the cardinality of the domain of its models (downward and upward versions of the above theorem); giving the paradox that set theory, whose main subject is the study of non-denumerable sets, does not even correctly determine this concept of denumerability. This is what is called Skolem’s Paradox (paradox, but it is not a contradiction, as he explained himself). And on this basis, Skolem established his “most important” result “that set theoretic notions are relative”.

Skolem’s paper included several other gems: remarks on the multivocity of the domain of sets (a fundamental fact of current set theory), on the reasons why he found it almost impossible to get a proof of consistency (still out of sight today), on the possibility that the Continuum Hypothesis was independent of ZFC (as it was later proved), etc.

In short, this is a truly fundamental article, in which he ended by saying: “Experts in set theory are usually of the opinion that the notion of integer must be defined and the principle of mathematical induction must be proved. But it is clear that we can neither define nor prove ad infinitum; sooner or later we will arrive at something that can no longer be defined or proved. So our only aim should be that the initial foundations are something immediately clear, natural, and not questionable. This condition is satisfied by the concept of the integer and by inductive inferences, but it is certainly not satisfied by conjunctive axioms of the Zermelo type, or anything of the sort. If we were to accept the reduction of those notions to the latter, set concepts should be simpler than mathematical induction, and reasoning with them should be less questionable, but the actual situation is entirely the opposite.”

To learn more, se the original work. English translation: Some remarks on axiomatized set theory, in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press.