4 removed contentious phrase "freely available"

Does there exist a set $A$ such that $A=\{A\}$ ?

Edit(Peter LL): Such sets are called Quine atoms.

Naive set theory By Paul Richard Halmos(freely available)Halmos On page three, the same question is asked.

Using the usual set notation, I tried to construct such a set: First with finite number of brackets and it turns out that after deleting those finite number of pairs of brackets, we circle back to the original question.

For instance, assuming $A=\{B\}$, we proceed as follows: $\{B\}=\{A\}\Leftrightarrow B=A \Leftrightarrow B=\{B\}$ which is equivalent to the original equation.

So the only remaining possibility is to have infinitely many pairs of brackets, but I can't make sense of such set. (Literally, such a set is both a subset and an element of itself. Further more, It can be shown that it is singleton.)

For some time, I thought this set is unique and corresponded to $\infty$ in some set-theoretic construction of naturals.

To recap, my question is whether this set exists and if so what "concrete" examples there are. (Maybe this set is axiomatically prevented from existing.)

3 spelling

Does there exist a set $A$ such that $A=\{A\}$ ?

Edit(Peter LL): Such sets are called Quine atoms.

Naive set theory By Paul Richard Halmos(feely avilableHalmos(freely available) On page three, the same question is asked.

Using the usual set notation, I tried to construct such a set: First with finite number of brackets and it turns out that after deleting those finite number of pairs of brackets, we circle back to the original question.

For instance, assuming $A=\{B\}$, we proceed as follows: $\{B\}=\{A\}\Leftrightarrow B=A \Leftrightarrow B=\{B\}$ which is equivalent to the original eqautionequation.

So the only remaining possibility is to have infinitely many pairs of brackets, but I can't make sense of such set. (Literally, such a set is both a subset and an element of itself. Further more, It can be shown that it is singleton.)

For some time, I thought this set is unique and corresponded to $\infty$ in some set-theoretic construction of naturals.

To recap, my question is whether this set exists and if so what "concrete" examples there are. (Maybe this set is axiomatically prevented from existing.)

2 standard name

Does there exist a set $A$ such that $A=\{A\}$ ?

Edit(Peter LL): Such sets are called Quine atoms.

Naive set theory By Paul Richard Halmos(feely avilable) On page three, the same question is asked.

Using the usual set notation, I tried to construct such a set: First with finite number of brackets and it turns out that after deleting those finite number of pairs of brackets, we circle back to the original question.

For instance, assuming $A=\{B\}$, we proceed as follows: $\{B\}=\{A\}\Leftrightarrow B=A \Leftrightarrow B=\{B\}$ which is equivalent to the original eqaution.

So the only remaining possibility is to have infinitely many pairs of brackets, but I can't make sense of such set. (Literally, such a set is both a subset and an element of itself. Further more, It can be shown that it is singleton.)

For some time, I thought this set is unique and corresponded to $\infty$ in some set-theoretic construction of naturals.

To recap, my question is whether this set exists and if so what "concrete" examples there are. (Maybe this set is axiomatically prevented from existing.)

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