Can you find a set $A$ which satisfies $A = \{2,|A|\}$?

Question

Can you find a set $A$ which satisfies $A = \{2,|A|\}$? Justify your answer.

Answer 1

3)If we do not allow multisets and do not allow sloppy notation, then whenever we have $M = \{x,y\}$ we know that $x \ne y$ and $|M| = 2$.

So if $A = \{2,|A|\}$ we know that $|A|\ne 2$ and $|A| = 2$. Which is a contradictions. So no such set exists.

I do not think this should be the answer because I believe $A= \{2,2\} =\{2\}$ and $|A|=1$ should be legitimate things to write.

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2)If we do not allow multisets (which we shouldn't) but allow sloppy notation (which in my opinion we should) and allow multiple representation of a single element-- for example; if $K=\{x^2\% 10|x\in \mathbb Z\}$ then $K=\{... (-3)^2\%10, (-2)^2\%10, (-1)^2\%10,0^2\%10, 1^2\%10, 2^2\%10, 3^2\%10, ....\} = \{..... 9,4,1,0,1,4,9,6,5,6,9,4,1,0,..\} = \{0,1,4,5,6,9\}$ and $|K|=6$..or for example $M=\{$letters in the work "construction"$\}=\{c,o,n,s,t,r,u,c,t,i,o,n\}=\{c,o,n,s,t,r,u,i\}$ and $|M|=8$---

Then if $M= \{x,y\}$ then $|M| = 2$ if $x \ne y$ and $|M| = 1$ if $x=y$.

So if $A= \{2,|A|\}$ we would have $|A| =1$ if $|A| = 2$ or we would have $|A|=2$ if $|A|\ne 2$. Both are contradictions and no such set can exists.

In my opinion this should be the answer.

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1) If we allow the word "set" to include "multisets" (which in my opinion we should not),

then $|\{2,2\}| = 2$. And also $|\{x,y|\}|= 2$ regardless as to whether $x=y$ or $x \ne y$. So if $A = \{2,|A|\}$ then $|A| = 2$ and thus we have $A = \{2,|A|\} = \{2,2\}$.

[Ordered sets are a form of multisets, so the $A= (2,2)$, the ordered pair, will also by such that $|(x,y)| = 2$ so $A=(2,2) = (2,|A|)$, will work]

In my opinion, these should not be the answers as "multisets" are not sets, and sets can not have multiple representations of elements. The set $\{2, |A|\}$, in my opinion, can be a set with two different elements, or a set with one distinct element that can be represent by being written twice. I think if you allow multisets and consider that a set with one element written twice, or in a distinct order to be a different object than a set with the one element written once, you are obligated to specify that.

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Answer 2

Hint: Note that as written, your set could have either one or two distinct elements, depending on whether $2$ and $|A|$ are equal or not. Try considering these cases separately.

Answer 3

If $A=2$, then $A=\{2\}$, a contradiction.

If $A\neq 2$, then $|A| = 2$, which means that in fact $A=\{2\}$ and so $|A|=1$, another contradiction.