Infinite Sets

Definition 45   A set S is infinite if and only if there is a monomorphism $M:S\to S$ and a point $p\in S$ which is not in the image of m.

Proposition 6.1   If A is infinite and $A\subseteq B$ then B is infinite.

Proposition 6.2   If S is infinite, then so is $S\times S$.

Proposition 6.3   If S is infinite, then so is S + S.

Proposition 6.4   If S is infinite, then so is ${\cal P}(S)$.

Proposition 6.5   If S is infinite, then there is a monomorphism $f:\ensuremath{\mathbb{N} }\to S$.

Proposition 6.6   If S is infinite, then so is S^S.

Proposition 6.7   is infinite.

Proposition 6.8   If $f:A\to B$ is an epimorphism and B is infinite, then A is infinite.

Proposition 6.9   If S is infinite then there is a strictly decreasing sequence of subsets of S:

$$S_0\supset S_1 \supset S_2 \ldots$$

Proposition 6.10   If S is infinite then there is a strictly increasing sequence of subsets of S:

$$S_0\subset S_1 \subset S_2 \ldots$$

Proposition 6.11   If $A\cup B$ is infinite, then either A is infinite or B is infinite.