In the book 'Practical Foundations of Mathematics' (Paul Taylor), available online, one reads:

As Emile Borel stressed in 1908, the important observation about $\mathbb{Q}$ [that there is a bijection from $\mathbb{Q}$ to $\mathbb{N}$] is that there is an effective coding, not anything to do with its "size".

Does anyone know where evidence for Borel stressing this can be found?

  • $\begingroup$ I think: E.Borel, Sur les principes de la théorie des ensembles, 1908 presumably reprinted as Note IV of the 2nd ed of Leçons sur la théorie des fonctions. But I cannot find it on the web... $\endgroup$ May 22, 2017 at 10:02
  • $\begingroup$ Some hints in Gregory Moore, Zermelo's Axiom of Choice: Its Origin Development and Influence, Springer (1982), page 102. After 1905 Borel became more skeptical about Cantor's theory of alephs. He thinked that the there were nor uncontable infinite sets but only effectively denumerable sets and denumerable sets that are not effectively so. If so, the observation above must be read as: the proof of the existing bijection between $\mathbb Q$ and $\mathbb N$ is interesting because it gives an effective way to "enumerate" $\mathbb Q$. $\endgroup$ May 22, 2017 at 11:45

1 Answer 1


This statement is characteristic of Borel's change of attitude in 1908, around the Fourth International Congress of Mathematicians. Borel always had constructivist leanings, and considered all infinities to be only potential (unfinished). This position became more pronounced after Zermelo's controversial proof of the well-ordering theorem from the axiom of choice appeared in 1905. But as late as 1908 Borel still identified allowable infinities with denumerable (countable) infinities, i.e. by size:

"...a non-denumerable infinity of choices (successive or simultaneous)... appears to me... entirely meaningless. As for a denumerable infinity of choices, clearly one cannot effect them all, but... one is assured that any given choice will be effected at the end of a finite time".

This changed in Les «Paradoxes» de la Théorie des Ensembles (1908). Denumerability was of course still necessary for meaningful infinities, but no longer sufficient. Borel introduces a stronger notion of effective enumerability, which will come into prominence two decades later. Here is from Moore's Zermelo's Axiom of Choice:

"In one respect Borel refined his earlier position. No longer did he regard the relevant distinction as that between denumerable and non-denumerable sets. From a practical point of view, the principal distinction was now between those sets effectively enumerable and those not. According to Borel, a set was effectively enumerable if one could state, "by means of a finite number of words, a definite process for attributing unambiguously a [unique natural number as] rank to each of its elements" [1908a, 446-447].

Some denumerable sets were not effectively enumerable, since to attribute a rank to each element might require an infinity of arbitrary choices. Moreover, an infinite subset of an effectively enumerable set need not be effectively enumerable. At last Borel had produced a positive concept, effective enumerability, in response to his qualms about set theory and Zermelo's Axiom. Related notions of effectivity for subsets of $\mathbb{N}$, such as recursive enumerability, were investigated by recursion theorists beginning in the 1930s."

I should note that Borel's attitude was rather pragmatic, he did not deny some indirect utility to non-constructivist mathematics, and even used Cantorian arguments in his work on complex functions and measure theory. But throughout his life he insisted on sharp separation between "Zermeloan" and non-"Zermeloan" mathematics, classing the former as only provisional. As he wrote to Hadamard in 1905:

"I prefer not to write alephs. Nevertheless, I willingly state arguments equivalent to those which you mention, without many illusions about their intrinsic value, but intending them to suggest other more serious arguments... One may wonder what is the real value of these arguments that I do not regard as absolutely valid but that still lead ultimately to effective results. In fact, it seems that if they were completely devoid of value they could not lead to anything since they would be meaningless collections of words. This, I believe, would be too harsh. They have a value analogous to theories in mathematical physics through which we do not claim to express reality but rather to have a guide that aids us, by analogy, in predicting new phenomena, which must then be verified."


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