the notion of generic in set theory

[José Manuel Rodríguez Caballero]

Dear FOM members,
  Concerning the following statement taken from [1]:

if you take set theory and you do not allow generic then obviously the
> continuum hypothesis is true, but if you allow generic, then obviously it
> is not true.


I would like to ask the following questions:
1) What does it mean generic in this setting?
2) Is there a formal definition of generic?
3) Which option, between allowing and not allowing generic, does mainstream
mathematicians prefer concerning set theory?

Kind regards,
Jose M.

 Reference:

[1] What is a Manifold? - Mikhail Gromov (time 31:28 / 53:55)

URL = https://youtu.be/u5DLpAqX4YA?t=1888
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