A simple question on an axiom in Set Theory

In the book of Set Theory, Thomas Jech mentions

Axiom Schema of Comprehensions (false): If $P$ is a property, then there exists a set $Y=\{x\,\,:P(x)\}$.

Then he mentions that this principle is false.

My question may not be so good, but as a curiosity, it came to my mind.

Why it has name (Axiom Schema of Comprehensions (false))? Is it a standard name to this axiom or named only by few authors in modern text?

The actual axiom of comprehension allows you to identify elements of an existing set that have any given property. They do not allow you to identify anything in the entire universe with a property.

Example: Once you have shown the natural numbers exist as a set, you can then show that any property about natural numbers, you can generate a set of the elements of the natural numbers with that property. (Odd, even, prime, etc.)

What you cannot do is name a property and then create a set out of nothing of things with that property, they have to have come from an existing source.

The reason for this is to prevent things that are "too big" to be a set, which would cause contradictions. Another is to avoid pathological sets like Russell's paradox. Let $P(x)$ be the property that $x\not \in x$. Now, just about every set would have this property, sets in general are not elements of themselves (In fact, in ZF set theory we prevent this kind of infinite descent with other axioms)

But let's say you allowed unlimited comprehension, so this set of sets that don't contain themselves exist, call that set $R$. Now, the question is, is $R\in R$? Well, if it is, it can't be, because by definition $R$ only contains things that don't contain themselves. But if it isn't, it must be, because that's the definition of $R$! So we get a breakdown of the law of the excluded middle, a statement that can neither be true nor false. This is what most mathematicians would call a Bad Thing, thus we prevent unlimited comprehension

Addition: Just as a note, the reason this is an axiom schema and not an axiom is it is in fact an infinite family of axioms, any property P(x) has its own specific axiom that allows you to comprehend that property.

Why the name?

It is called the axiom schema of comprehension because it says, informally, that for any property you can think of, i.e. comprehend, there exists a set containing all objects that have have that property.

Is it a standard name?

It is a standard name but it has other names too. Some authors call it the "axiom schema of naïve comprehension" or full comprehension. Naïve comprehension is the more common name. The word naïve is used because any set theory which takes it as an axiom is inconsistent for reasons pointed out by @Alan. The word full is used because the axiom places no restriction on which properties we can use to construct new sets. Hence, another name is "axiom schema of unrestricted comprehension". I want to stress that I am not asserting what the axiom says. The axiom is false in ZFC.

Don't get confused by different names.

The axiom that Jech calls "axiom schema of separation" is sometimes called axiom schema of comprehension" or "axiom schema of restricted comprehension" (less commonly, the subset axiom scheme). This is not the comprehension that Jech means but it is the comprehension that Alan means. The thing to remember is that one axiom schema says that for any property, there exists a set that contains all objects with that property. This one is inconsistent and false in ZFC. The other axiom schema adds the requirement that the set must be a subset of some other set which already exists. This one is true in ZFC.