Among several possible definitions of ordered pairs - see below - I find Kuratowski's the least compelling: its membership graph (2) has one node more than necessary (compared to (1)), it is not as "symmetric" as possible (compared to (3) and (4)), and it is not as "intuitive" as (4) - which capt...
Proposition:
Each subset of the natural numbers is finite or countable.
Proof:
Let $X \subset \omega$.
First case: $X$ is bounded. That means that $(\exists k \in \omega)(\forall y \in X) y \leq k$. Then $X \subset k+1$ and $X$, as a subset of a finite subset, is finite .
Second case: $X$ i...
Proposition:
Each subset of the natural numbers is finite or countable.
Proof:
Let $X \subset \omega$.
First case: $X$ is bounded. That means that $(\exists k \in \omega)(\forall y \in X) y \leq k$. Then $X \subset k+1$ and $X$, as a subset of a finite subset, is finite .
Second case: $X$ i...
