How elements are defined in axiomatic set theory I'm trying to understand the axioms of axiomatic set theory. I'm studying this book and I didn't understand how can we define the elements of a set and the set $\{x\}$. If I define the singleton, I know how to define the finite sets using the union axiom.

Afterwards, the author says

The author just says: "given any $x$ we have the singleton $\{x\}$", which axiom he used to say that? afterwards the author says the definition of $\{x\}$, which in my opinion is wrong because using (I), $\{x\}$ should be define as $\{\{x\},\{x\}\}$.
I need help
Thanks in advance
 A: Uses the pair set axiom $\{x\}$ is by definition $\{x,x\}$.
"For any sets $u$ and $v$ ($x$ and $x$) the pair set $\{u,v\}$ ($\{x,x\}$) is the set...".
A: In the Pairing Axiom, there is nothing requiring the variables $u$ and $v$ to take distinct values.  In the case $u=v$, the resulting set $B$ is a singleton set.
A: In axiomatic set theory we don't "define elements". Everything1 is an element, and everything is a set. There is no issue with sets being elements of other sets, if you consider the power set of $A$, it's a set and all its elements are sets themselves.
Axiomatic set theory says what are the properties of the $\in$ relation, and what sets we can prove to exist (given the existence of some sets).
For example, the axiom of pairing says that if $x$ and $y$ exist, then the set $\{x,y\}$ exists. Since we do not require that $x$ and $y$ are distinct, if $x=y$ we have that $\{x,x\}$ exists. But since repetition is discarded when considering sets, we have that $\{x,x\}=\{x\}$.
It might be strange, at first, to talk about "proving that $\{x,y\}$ exists", because when we first start with mathematics we are used to be given a world, and work in that world. The real numbers are given, we don't have to prove that $\sqrt2$ or $\pi$ exist, of course they do. There's no question.
But in logic, and in set theory, we transcend to a more abstract universe of mathematics, a universe which are we not familiar with, and can't fully imagine. And in that universe all we have is the axioms of set theory and $\in$, to light up our way as we trudge through the sets. So if you want to argue that there is a particular set, e.g. $\{x\}$, in the universe of set theory, then you have to prove that from your axioms.
Even if it seems "obvious", and even if it seems "preposterous". All that matters is that we can prove or disprove things. Luckily the axioms were chosen to be very compatible with our intuition when it comes to basic things like the existence of $\{x\}$, or the power set of $x$, and so on.

Footnotes.


*

*There are set theories in which we allow the existence of proper classes, there sets are defined as classes which are elements of other classes.
There are set theories in which we allow the existence of atoms, or urelements, which are not sets, there sets are defined as objects which have elements (and the empty set as well).
A: Let me take a crack at this.
So let us start with what the author says.
The author (in two sentences) says this:  We can use pairing and union together to form other finite sets.  First of all, given any $x$, we have the singleton $\{x\}$, which is defined to be $\{x,\ x\}$.
Now your question: What axiom(s) is the author is using to say the first of the two sentences?
Let me paraphrase what the author says:
Here's the Paraphrase of the first sentence:  We can use the Pairing Axiom together with the Union Axiom to construct other sets---finite ones!
The author continues...
Here's the paraphrase of the second sentence:  Given any set $x$, we can use the Pairing Axiom to contruct the set $\{x,\ x\}$.
It is now clear that the author is applying the Pairing Axiom to a pair of identical sets $u$ and $v$ which he just calls $x$.  There is nothing in the Pairing axiom which says that the given sets $u$ and $v$ had to be different nor does it say that have to be the same.  So if $u$ and $v$ are the same (or identical) sets, then the Pairing Axiom allows us to construct the following identical sets $$\{u,\ v\},\ \{v,\ u\},\ \{u,\ u\},\ \{v,\ v\}$$The author then defines the newly constructed pair sets as a singleton which would be $\{u\}$ or $\{v\}$. In your case, we have $\{u\} =\{v\} = \{x\}$  and not $\{\{x\},\ \{x\}\}$ as you thought originally.
Note: The set variable $x$ represents a set which could be empty or nonempty.
On the other hand, if the given sets $u$ and $v$ were not identical (or not the same), then the Pairing Axiom allows us to construct the  identical pair sets $\{u,\ v\}$ and $\{v,\ u\}$ which has two distinct elements $u$ and $v$.  In this case $\{u,\ v\} = \{v,\ u\}$ is called a Doubleton.
The author goes on to say how sets containing a finite number of elements can be constructed using the Pairing and Union Axioms.   Hence, we now
can construct finite sets such as
$$\{\emptyset\},\ \{\emptyset,\ \{\emptyset\}\},\ \{0,\ 1,\ 2,\ 3,\dots, 12\},\ \{a,\ e\}\ \cup\ \{i,\ i\}\ \cup\ \{o,\, u,\ o,\ u\} = \{a,\ e,\ i,\ o,\ u\},\dots$$
Note:  A singleton in the Theory of Sets is any nonempty set containing a single element (or just one element).
A finite set such as $\{5, 5, 5, 5\}$ is thought to contain one element---namely $5$ in this case and so
$$\{5,\ 5,\ 5,\ 5\} = \{5\}\ \ \hbox{is a singleton}$$
On the other hand, $\{\{5\},\ \{5\}\} = \{\{5\}\}$ is also a singleton whose only element is also a set---namely, the singleton $\{5\}$.
Please note the difference
$$\{5\}\ \neq\ \{\{5\}\}\ \ \ (\hbox{because}\ \ 5 \neq\ \{5\})$$
I hope this explanation clears up the confusion.
