Georges Reeb's "claim Q" is the statement that "naive integers don't fill up $\mathbb{N}$". To anyone familiar with model theory this could easily be interpreted as the existence of nonstandard models of the natural numbers, whether in PA or ZF, known since (Skolem 1933):
Skolem, T. "Über die Unmöglichkeit einer vollständigen Charakterisierung der Zahlenreihe mittels eines endlichen Axiomensystems." Norsk Mat. Forenings Skr., II. Ser. No. 1/12 (1933), 73--82.
Reeb published two articles with an identical title:
Reeb, G. "La mathématique non standard vieille de soixante ans?" Publications de l'Institut de Recherche Mathématique avancée 1979 64/P-30, Université de Strasbourg, 1979
and
Reeb, Georges. "La mathématique non standard vieille de soixante ans?" (French) [Nonstandard mathematics sixty years old?] Third Colloquium on Categories, Part III (Amiens, 1980). Cahiers Topologie Géom. Différentielle 22 (1981), no. 2, 149-154.
Even though the titles are identical, Reeb presents different arguments for his "claim Q" here.
In the 1979 article Reeb takes a more "fundamental" attitude and seems to argue for his "claim Q" somehow "from first principles", the naive integers taken to exist before a commitment to formal mathematics. Thus, Reeb seems to think naive integers don't fill out $\mathbb{N}$ but he doesn't provide much of an explanation of this. One comment that he does make is that being naive is not a mathematical concept; hence it not formalizable; hence they cannot be said to fill up $\mathbb{N}$.
However, to many mathematicians this would sound no more convincing than a claim that a naive concept of a finite graph (or another concept from finite combinatorics) could not fill up the formalized mathematical concept. The justification would be the same but the claim is apparently less convincing than in the case of the integers which are somehow more open-ended at infinity.
In the 1981 article in Cahiers Topologie Géom. Différentielle Reeb takes a different tack. Here the approach seems to be that any construction in mathematics, as recognized since the time of Hilbert, necessarily introduces "ideal" elements not intended by initial naive intuitions. (This approach is closer to the nonstandard model argument.)
There are Nelson-style justifications for "claim Q" but they don't seem to be the same as Reeb's, who was apparently not a finitist at this stage in his career. There are various motivations for why naive integers shouldn't exhaust $\mathbb{N}$, my favorite one being about the number that's too big to be expressed by even the computer the size of the universe running the entire time allotted to our civilisation and exploiting the fastest growing functions in our logical arsenal; such numbers would apparently function as infinite for all practical purposes at the "naive" level. This indicates a lack of homogeneity of $\mathbb{N}$ which can serve as an argument in favor of enriched syntax as in Nelson's system, which singles out "standard" (or "assignable") elements out of $\mathbb{N}$ by means of a single-place predicate (violating the separation axiom).
Is it really true that there is nothing of this sort in Reeb himself, and what is ultimately his justification for "claim Q"?
Note. The issue is dealt with in more detail in this recent article.
