I’m supposed to prove that natural number $n$ cannot be an ordered pair. The definition for ordered pair is $(x,y) = \{\{x\},\{x,y\}\}$ and for the definition of natural numbers we use the definitions of set theory.
I don’t know if this should be done with induction or what. Any clues are helpful. Thanks!
Note that an ordered pair has between $1$ and $2$ elements. This limits the number of cases to exactly $1$ and $2$.
Or, you can note that $\varnothing$ is never an element of an ordered pair, and an ordered pair is never empty.
Note that if you are using Zermelo's natural numbers, then $2=\{1\}=\{\{0\}\}=\{\{\varnothing\}\}=\langle\varnothing,\varnothing\rangle$. But the above assumes that you're working with the standard von Neumann definition where $n=\{0,\dots,n-1\}$.