At 31:37 in his lecture titled *[What is a manifold?](https://youtu.be/u5DLpAqX4YA?t=31m37s)* posted on Youtube, Mikhail Gromov states that if we do not allow generic functions to exist then the continuum hypothesis is "obviously" true, and that if we do allow generic functions to exist, then the continuum hypothesis is "obviously" false.


**What concept is he referring to?** It sounds extremely important.


Note: this is a follow-up to a question I posted on Math.SE: <https://math.stackexchange.com/questions/1887350/what-is-a-generic-genetic-geometric-map-in-the-study-of-manifolds>


Also I'm not sure how to tag this question, so please feel free to fix the tags as appropriate.


**EDIT:** When Gromov mentions generic functions for the first time in the lecture, around 29:30 if I remember correctly, it seems like it is with regards to generic points (which are generic by Sard's Theorem I believe) for which the Implicit Function theorem can be used to generate a manifold from the equation $f(x)=0$; the functions $f$ for which this holds are what he refers to as "generic". I don't know if this helps anyone at all; I don't really understand Gromov's terminology.



Does anyone understand this comment about the continuum hypothesis?
I remember reading in Proof from the Book (starting p. 119) that Erdős proved the equivalence between the continuum hypothesis and the following problem :
Let $(f\_\alpha)$ be a pairwise distinct family of analytic functions over $\mathbb{C}$, sur that for any $z\in \mathbb{C}, (f\_\alpha(z))$ is countable. Is the family $f\_\alpha$ is itself countable?
(Edit : more precisely, $c > \aleph\_1$ iff any such family is countable, otherwise you have such a family with cardinality $c$).
It looks like a good way to interpret the CH towards an explication of Gromov's statement.


