The notion that the union of two finite sets is again finite is something I took as intuitively true for quite a while. A proof using arithmetic is relatively straight forward.
Suppose $A$ and $B$ are finite sets, so let $m=|A|$ and $n=|B|$. Now $A\cup B=A\cup(B-A)$, and since $B-A$ is a subset of a finite set, it is also finite (I have already seen this statement proven), so let $k=|B-A|$. Then $|A\cup B|=m+k=m+_\omega k\in\omega$, so $A\cup B\approx m+_\omega k$, and thus finite.
I've read that it is possible to prove this without any use of arithmetic, but I don't quite see how, as arithmetic seems too essential in showing a set is equinumerous to a natural number. Does anyone have a reference to or a reproduction of such a proof?, I'm curious to see. Thank you.
Edit: I was unaware of other definitions of what it means to be finite. I consider a set $A$ finite if there is a bijection $f\colon A\to n$ for some natural number $n$. For arithmetic, I'm trying to refrain from using cardinal arithmetic or the arithmetic of natural numbers, so no $+_\omega$, as was used in the other proof I gave. Also, my definition of natural number is the recursive definition that $0=\emptyset$, and $n^+=n\cup\{n\}$.