Origin of the term "generic" in set theory In set theory, in particular the context of forcing, if $M$ is a model of $\sf ZFC$ and $P\in M$ is a partial order, we say that $G\subseteq P$ is a generic filter (or $M$-generic or generic over $M$) if:


*

*Whenever $p\in G$, and $p\leq q$ then $q\in G$.

*Whenever $p,q\in G$ then there is some $r\in G$ such that $r\leq p,q$.

*If $D\subseteq P$ is dense (for all $p\in P$ there is $q\in D$ such that $q\leq p$), and $D\in M$, then $D\cap G\neq\varnothing$.


The first two conditions specify that $G$ is a filter, and the third is the required genericity. So a filter is generic if it meets all the dense sets in the ground model. Since this context allows it, we can replace dense by "open dense", meaning $D$ is dense and if $p\in D$ and $q\leq p$ then $q\in D$.
So being generic means meeting all the dense open sets. And this definition agrees with the definition of genericity in the contexts of topology and algebraic geometry (and perhaps other fields of mathematics as well).

Question. Where did the term "generic" come from originally to mathematics, and how did it trickle into set theory?

It should be noted that Cohen used "generic" in his original paper, but it seems to have a different meaning, and not quite this one.
 A: In trying to trace the history of forcing in an earlier MO question, I came across G.H. Moore's The origins of forcing. I think you can find in Moore's piece an answer to your question, too. On p. 164 he writes:

From the corresponding paper of Solovay, A Model of Set-Theory in which Every Set of Reals is Lebesgue Measurable, p. 4:

The next page footnotes:

Our original definition of generic was based on "complete sequences". The present approach is due to Levy [8].

I have found no copies of the Levy papers, but Solovay's citations of 8 and 9, respectively, are:

A: As far as the usage in set theory is concerned, Cohen gives the following explanation in his book, Set Theory and the Continuum Hypothesis, after explaining a failed attempt to construct a suitable set $a$:

Rather than describe $a$ directly,
  it is better to examine the various properties of $a$
  and determine which are desirable and which are not.
  The chief point is that we do not wish $a$ to contain
  "special" information about $M$,
  which can only be seen from the outside …
  The $a$ which we construct will be referred to as
  a "generic" set relative to $M$.
  The idea is that all the properties of $a$ must be "forced"
  to hold merely on the basis that $a$ behaves like a "generic"
  set in $M$.
  This concept of deciding when a statement about $a$
  is "forced" to hold is the key point of the construction.

Certainly Cohen was partially motivated by the prior use of the term in geometry and topology, and in particular the fact that generic sets should satisfy any condition that is "dense."
