Difficulty in reading Introduction to Set Theory by Hrbacek Background: I was reading Karel Hrbacek's Introduction to Set Theory and need help with understanding this snippet:

What I don't understand/need to clarify:

*

*In order to be allowed to use the notion $\{x| P(x)\}$ I need to prove that: $\exists A:\forall x:\big(P(x)\implies x\in A\big)\implies \{x\in A|P(x)\}\textit{ exists and it's unique}$, right?

*I've noticed that sometimes it's not mandatory to prove the existence and uniqueness of an object to define it. For instance: When defined what $A\subseteq B$ mean for two given sets $A$ and $B$. So, when is it mandatory to prove the existence and uniqueness of an object in order to define it?
Update:


*Why is it even mandatory to prove the existence and uniqueness of an object to define it?

 A: *

*You are quite correct. The famous example is the set $\{x \mid x \not\in x\}$ that stars in Russell's paradox. You can't use that notation to describe a set (in ZF set theory), because you can't prove the existence of a set $A$, such that $x \not\in x \Rightarrow x \in A$.

*Defining a relation between objects does not require you to construct an object that represents that relation. The set of all pairs of sets $(A, B)$ such that $A \subseteq B$ isn't a set in ZF set theory, but that doesn't invalidate the definition of the subset relation. I would describe introduction of new relations as definitional.

*This is about extending a theory $T$ by adding a new constant symbol $c$ that satisfies an axiom $P(c)$ for some formula $P$ (or maybe, mutatis mutandis, a new function symbol $f$). Such an extension is consistent if $\exists x\lnot P(x)$ cannot be proved in $T$. Such an extension is conservative if adding the axiom $P(c)$ to $T$ doesn't introduce any theorems that do not involve $c$ and were not already theorems of $T$. Such an extension is definitional (my terminology) if $T$ proves $\exists!x(P(x))$. Definitional extensions are conservative and conservative extensions are consistent.

If an extension is not consistent (i.e., if $P(x)$ leads to a contradiction, then the resulting theory is trivial (everything is true). Apart from that, it is a judgment call whether you require extensions to be conservative or definitional. The usual presentations of ZF set theory comprises a sequence of extensions to the theory of a binary relation $\in$ that are either definitional or not conservative.
A: We have really great and important questions here. First, let’s see what exactly Hrbacek and Jech are saying in their (really good) book.
As you may know, the Axioma Schema of Comprehension states that

Let $P(x)$ be a property of $x$. For any set $A$ there is a set $B$ such that $x \in B$ if and only if $x \in A$ and $P(x)$.

Informally, this axiom says that, given any set $A$, we can always “seperate” those elements of $A$ that satisfy the property $P(x)$ and collect them to form a new set $B$. So far, so good.
As the authors points out, this set ($B$) is uniquely determined, meaning, any other set $C$ in these conditions will be equal to $B$. Now, we are completely certain that a set in these conditions exists and is unique. Therefore, we can assign them some terminology or notation.
Since this set is uniquely determined, any time we use the notation $\{x \in B \mid P(x)\}$ we all know what set we are talking about (because it is unique, and therefore this notation is not ambiguous).
For the last part of the text from the book, are you familiar with classes? Although it is not an object of the standard universe of sets, some theories regard this as a proper object. Informally, a class is a collection of objects that fails in being a set by being too large. Take for example the class of all sets. Sometimes, we denote this “collection” by $\{x \mid x = x\}$. But we don’t want, with this notation, to say that this collection is a set. In fact, one can prove using the ZF axioms that there is no such set.
You may now ask, why did they allow this later notation for sets? Well, they are not wrong, since they advertise that the use of this, let’s call it an abuse of notation, must only occur when we are sure that there is a set consisting of those elements. For example, let $A$ and $B$ be any sets. We know that $\{x \mid x \in A \vee x \in B\}$ is a set. One may argue that something is missing, namely when we first write $x$ (as one sees in the axiom schema of comprehension). Although, one can prove that there is a set $X$ such that $x \in X$ if and only if $x \in A$ or $x \in B$, so that use of notation is legit.
Now, for the questions 1-3.

*

*I believe I already answer this above. When we are defining a new object, we must ensure that the object exists (so when we are talking about it we are actually talking about something that exists) and that is unique (in this way, when we talk about it, we all know what we are talking about, and there is no risk of confusion or ambiguity).


*Although no one proves the existence of such relation, be aware that $A \subseteq B$ is a shorthand notation for the formula $\forall x (x \in A \Rightarrow x \in B)$. On the other hand, considering $\subseteq$ as a binary relation one can prove that this relation exists (it is proving that some set exists, since a binary relation is a set itself). I believe that the reasons that no one do this is that the proof itself is not hard, it follows directly by applying and manipulating some ZF axioms (note that if we were to do such proves all the time, we wouldn’t go anywhere) and, finally, it is such an intuitive result that one would have no big trouble in accepting it.


*For this, read again what I wrote in 1.
Remark. For the second point, I don’t want to encourage you or anybody to accept results without a proof or, at least, without asking why. This question of yours is really essential, because it is good for a mathematician to have these doubts and to think of these issues. What I meant is that, with time, and with practise we come to accept this results, because deeply we now that a proof is possible and we would be able to do it (with more or less effort).
I hope this helps you with your study.
Second Remark. On the page $9$, the Axiom Schema of Comprehension is introduced. This axiom tells us that, given a certain set, let’s say $A$, we can collect those elements of $A$ which satisfy some property and collect them to form a new set. On the next page, there is a Lemma which guarantees that this new set is unique. Therefore, given a set $A$, there is a unique set whose elements are those of $A$ which satisfy the property, let’s say $P(x)$. This set, which is uniquely determined, is denoted by $\{x \in A \colon P(x)\}$. In this notation, we first specify where the elements of these sets come from (we write $x \in A$, which means that its elements come from the set $A$), and then we specify the property that they satisfy (we then write $P(x)$).
Later, on page $11$, the author talks indirectly about classes. A class is a collection of elements that satisfy some property $Q$ and we denote classes by $\{x \colon Q(x)\}$. Not every class is a set, but every set is a class. A class is, so to speak, a collection of elements. And intuitively we know that a set is also a collection of some elements. Although, not every class is a set. For example, we may have the class of all sets, which is denoted by $\{x \colon x = x\}$. Since every set is equal to itself, then every set belong to this class. But there is no such set. Hence, when we write $x \in \{x \colon x = x\}$ we are actually just saying that $x$ is a set.
Although, sometimes we find the notation $\{x \colon Q(x)\}$ for sets. For example, $\mathbb{R}^2 = \{(x,y) \colon x,y \in \mathbb{R}\}$. In order to use this notation, we have to make sure that this is well-defined. [Later, in your study of ZF theory, you will understand why $\mathbb{R}^2$ is a set.] This actually means that we can gather all those object of the form $(x,y)$ to form a new set. And this set that we form is unique (that’s why that author says that it doesn’t depend on the set $A$, and for any set $A’$ constructed in a similar way will be the same as $A$). Hence, this set is uniquely determined and we can associate a notation for them (which is $\{x \colon P(x)\}$). To sum up, if we can collect objects that satisfy a property $P(x)$ in order to creat an actual set, then this set is unique.
