As with introducing Equivalent Sets, we want to talk about the size of a set.^{1}

A finite set is one that is equivalent to some natural number; an infinite set is one which is not finite. From this we can show that $\omega $ is infinite. This justifies the language “principle of infinity” with Natural Numbers. The principle of infinity asserts the existence of a particular infinite set; namely $\omega $.

It happens that if a set is equivalent to a natural number, it is equivalent to only one natural number.

A set can be equivalent to at most one
natural number.^{2}

A consequence is that a finite set is never equivalent to a proper subset of itself. So long as we are considering finite sets, a piece (subset) is always less than than the whole (original set).

A finite set is never equivalent to a proper
subset of itself.

Every subset of a natural number is equivalent
to a natural number.^{3}
A consequence is:

Every subset of a finite set is finite.^{4}

If $A$ and $B$ are finite, then $A \union B$
is finite.

If $A$ and $B$ are finite, then $A \times B$
is finite.

If $A$ is finite then $\powerset{A}$ is finite.

If $A$ and $B$ are finite, then $A^B$ is
finite.

- Will be expanded in future editions. ↩︎
- Future edition will include proof, which uses comparability of numbers and the results of Equivalent Sets. ↩︎
- This requires proof, and may become a proposition in future editions. ↩︎
- An account will appear in future editions. ↩︎