TITLE: What did Paul Cohen mean by saying that generic sets of natural numbers have "no asymptotic density?"
QUESTION [12 upvotes]: In Paul Cohen's original 1963 paper on forcing, The independence of the Continuum Hypothesis, published in PNAS, he gives his general proof sketch of how he intends to create a model of ZFC that doesn't support CH:

Start with a countable model of ZFC.
Within the model, there are the model's cardinals $\aleph_0^*, \aleph_1^*, \aleph_2^*,$ etc, which are different from the cardinals in the ambient theory $\aleph_0, \aleph_1, $ etc.
Cohen then proceeds building an ordinal-length sequence of subsets of the true $\aleph_0$ which is of length $\aleph_2^*$, and uses this to build a new model in which CH doesn't hold.

Of these subsets, Cohen writes:

"Only those properties which are true in a "uniform" manner for "generic" subsets of $\omega$ in $\mathcal{M}$ [the first model] shall be true for the $a_\delta$ [the ordinal-length sequence of subsets] in $\mathcal{N}$ [the new model]. For example, each $a_\delta$ contains infinitely many primes, has no asymptotic density, etc."

I interpret "no asymptotic density" as meaning the natural density doesn't converge to anything as $N \to \infty$.
I am not sure what Cohen means with the language "generic," but if the most straightforward interpretation is that it means "almost all," then this doesn't seem like a true statement for "generic subsets" of $\Bbb N$, as almost all have natural density equal to 0.5. If identify any such subset with its characteristic function from $\mathbb N \to \{0, 1\}$, then you have an infinite length binary string, and it would seem that almost all of those have 50% 0's and 50% 1's, similarly to how almost all real numbers have 50% of their bits equal to 0 and 50% equal to 1 in their binary expansion.
Does anyone have clarification on this? Is this a simple error or is there a different interpretation here of a "generic" subset?

REPLY [6 votes]: To compliment the comments and Nate Eldredge's answer, here's a useful lemma for showing that a given property $P$ of infinite binary sequences is comeager:

Lemma: Consider the game $G_P$ where players $1$ and $2$ alternate playing finite binary strings for $\omega$-many moves, and then player $2$ wins iff the concatenation of those strings has property $P$. Then the following are equivalent:

Player $2$ has a winning strategy in $G_P$.

The set of $f$ with property $P$ is comeager in $2^\omega$.



So for example:

The set of sequences with infinitely many $0$s is comeager: just have player $2$ always play "$0$."

The set of sequences with arbitrarily long strings of $1$s is comeager: just have player $2$ play a length-$i$ string of $1$s on their $i$th move.


And, as desired:

The set of sequences with undefined asymptotic density is comeager: have player $2$ alternately play "really long" strings of $0$s or of $1$s. Note that here we have to take the opponent's behavior into account in a meaningful way, since the right notion of "really long" depends on the total length of the sequence built so far, not just how many moves have been played.

In my experience it's much easier to think in terms of winning strategies than in terms of (co)meagerness per se. Note that while I've phrased the lemma for the specific context of $2^\omega$, it holds much more generally (and so we can use the same intuition when working with wildly different forcings).