Why are the Separation axioms 'too weak to develop set theory with its usual operations and constructions'?

Question

I was reading Jech's Set theory; there after introducing Russell's Paradox, he asserted:

The safe way to eliminate paradoxes of this type is to abandon the Schema of Comprehension and keep its weak version, the Schema of Separation...

Once we give up the full Comprehension Schema, Russell’s Paradox is no longer a threat...

Replacing full Comprehension by Separation presents us with a new problem. The Separation Axioms are too weak to develop set theory with its usual operations and constructions. Notably, these axioms are not sufficient to prove that, e.g., the union $X \cup Y$ of two sets exists, or to define the notion of a real number.

Why did he say the Separation axioms are too weak 'to develop set theory with its usual operations and constructions'?

Maybe it might be trivial, but I'm not able to comprehend the point. Can anyone shed some light on the author's statement?

Answer 1

I think he just means that simply replacing Full Comprehension with Restricted Comprehension is not enough. Given sets $X$ and $Y$, you could infer the existence their union $X\cup Y$ using Full Comprehension alone. You cannot do that with Restricted Comprehension, so you would also need an axiom of unions (as in ZFC).

Answer 2

Suppose that you work with the following naive theory:

1. Extensionality, so you can say when two sets are equal;
2. Comprehension, so you can say that if you can describe a collection, then it is a set; and
3. Choice, if you fancy doing mathematics with the axiom of choice.

Comprehension does not restrict what it means to be a property. So it includes things like "$x$ is a subset of $A$", which gives you the power set axiom; it includes "$x$ is a member of a member of $A$", which gives you union, and so on and so forth.

But oh no, Comprehension is inconsistent. So we replace it with its weaker counterpart, Separation. But now we cannot guarantee the existence of power sets, pairs, unions, and the nice things we had before. Which is why we need to include them in our axioms. To see that, simply note that $\{\varnothing\}$ is a model of Extensionality+Separation+Choice. But not a model of Power. And $\{\varnothing,\{\varnothing\}\}$ is also a model of Extensionality+Separation+Choice, but it is not a model of Pairing.

And you can consider this as an exercise to try and push this further and construct a model of Extensinoality+Separation+Choice, where some $X$ and $Y$ exist in the model, but $X\cup Y$ does not.