the notion of generic in set theory

[Joe Shipman]

A generic object is one with no specific properties whatsoever. Anything definable cannot be generic. Cohen constructed a model where CH was false by taking a minimal model where it was true, and then adding enough ”generic sets“ of integers to falsify CH in a carefully organized way to avoid accidentally causing any statements he needed to be true to become false.

— JS

Sent from my iPhone

> On Feb 15, 2020, at 6:48 PM, José Manuel Rodríguez Caballero <josephcmac at gmail.com> wrote:
> 
> 
> 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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