I'm a beginner in set-theory, and I got to read:
"The collection of all sets satisfying a given property may not together form a set"
I would like to ask, could this statement be translated as follows (in general):
"The collection of all sets satisfying a given property may not together form a set that also satisfies the given property"
Or is the added qualification not needed ? --- meaning that the collection may form "something" that does not follow from the Axioms of Set Theory ?
On
The added modification makes the statement weaker. Let $P$ denote the given property. The original statement asserts that the collection of all sets satisfying $P$ is not a set at all. Whereas your modification asserts that the collection of all sets satisfying $P$ either fails to be a set (as in the original statement) or is a set, albeit one failing to satisfy $P$.
Note that the original version implies your modified version. However, by Russel's paradox the original version is true. Therefore, the possibility that you have added - that the collection of sets satisfying $P$ is a set failing to satisfy $P$ - is unnecessary.
Remark: In set theory, a collection that is not a set is called a proper class. Therefore, another way to rephrase the original statement is
"The collection of all sets satisfying a given property may be a proper class."
Its actually stronger than that. The collection of all sets that satisfy a certain property may form what is known as a proper class. A proper class is a collection of sets which is to big to be a set itself. Which means ZF can prove that it does not exist. (ZF can prove that no set exists such that this set contains all sets which satisfy the aforementioned property)
For example, if we apply this to the property, x=x, then the collection of all sets which satisfy this property, is just the proper class containing all sets. ZFC can prove that a set containing all sets cannot exist, because sets are not allowed to contain themselves in ZFC, due to the axiom of foundations.
This is slightly informal, but every time you have the collection of all sets satisfying $\varphi(x)$, is not a set itself, this will happen because you can show that the "number" of sets which satisfy that property is the same size as the total "number" of sets. (calling it a number is very very much problematic, but I think its a useful way to think about it)