Countable Sets

Definition 48   A set S is countable if either $S=\emptyset$ or there is an epimorphism $f:\ensuremath{\mathbb{N} }\to S$.

Theorem 6.20   A set S is countable if and only if there is a monomorphism $m:S\to \ensuremath{\mathbb{N} }$.

Proposition 6.21   The finite cardinal set [n] is countable for any $n\in \ensuremath{\mathbb{N} }$.

Proposition 6.22   is countable.

Proposition 6.23   If A and B are countable then so are $A\times B$, A + B, and $A\cup B$.

Proposition 6.24   If $\Lambda$ is a countable set and $A_\lambda$ is countable for each $\lambda\in \Lambda$ then $${\bigcup_{\lambda\in \Lambda}A_\lambda}$$ is countable. (Any countable union of countable sets is countable.)

Definition 49   A number r is called algebraic if it is the root of an equation with integer coefficients. A real number which is not algebraic is called transcendental.

Proposition 6.25   There are countably many algebraic real numbers.