Set Theory – Russell’s Paradox and Set Definition in Terry Tao’s Analysis I

Question

In his book "Analysis 1", Terry Tao writes (check out page 39):

To summarize so far, among all the objects studied in mathematics, some of the objects happen to be sets; and if $x$ is an object
and $A$ is a set, then either $x\in A$ is true or $x\in A$ is false. (If $A$ is not a set, we leave the statement $x\in A$ undefined; for instance,
we consider the statement $3\in 4$ to neither be true or false, but
simply meaningless, since $4$ is not a set.)

But when discussing Russell's Paradox, he defines on page 53 a set

$\Omega := \{x : x \text{ is a set and }x\notin x\}$.

So he defines that an arbitrary object $x$ is an element of $\Omega$ if and only if $x$ is a set and $x\not\in x$. But this definition does not make any sense, since, according to his definition, we would have
$4\in\Omega$ if and only if $4$ is a set and $4\not\in 4$. But $4\not\in 4$ is meaningless, as he says, and therefore "$4$ is a set and $4\not\in 4$" is meaningless as well.

QUESTION:

How to fix this fault?

Note: I understand Russell's Paradox. But the definition

$\Omega := \{x : x \text{ is a set and }x\not\in x\}$

does not satisfy me formally.
My question is exactly how to make it formally work.

Answer 1

I actually asked this question Terry Tao himself. Half an hour ago, I got the following response per email:

A conjunction such as "A and B" is false when A is false, even if B is undefined. For instance, the assertion "x is non-zero and y = 1/x" is false when x=0, even though 1/0 is undefined. (Similar to the concept of lazy evaluation https://en.wikipedia.org/wiki/Lazy_evaluation in programming, one does not need to evaluate all inputs to a statement if one can already determine the truth of that statement from the portion that one is able to evaluate.)

Best,

Terry