Im having trouble with a homework question from my Set Theory class. The question is, Let $n$ be a set and an element of the natural numbers. If $x$ is an element of $n$, is $x$ also an element of the natural numbers? My intuition says yes, since we built up the natural numbers using only the empty set, I would guess that $x$ is just $n-1$, but I'm having trouble proving this.
My class is using the Zermelo-Fraenkel Set Theory axioms.
Assuming the von Neumann construction of natural numbers, the answer is yes.
Since $n=\{0,1,\dots,n-1\}$, an element of $n$ is also an element of $\Bbb N$.
In the special case $n=0$ (that is, $n=\emptyset$) it is vacuously true as no set is an element of $n$.
With Zermelo's construction, the answer is also yes.
Here, $n=\{\{\dots\}\}$, where there are $n+1$ nested brackets. So except for the case $n=0$ (which is still vacuously true), $n$ has only one element, which is $n-1$, so any element of $n$ is a natural number.