Return to Question

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) 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.)

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 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.)

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.)

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) 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.)

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

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.)

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.)