Infinite

Definition 62   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 137   If $A$ is infinite and $A\subseteq B$ then $B$ is infinite.

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Proposition 138   If $S$ is infinite, then so is $S\times S$.

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Proposition 139   If $S$ is infinite, then so is $S + S$.

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Proposition 140   If $S$ is infinite, then so is ${\cal P}(S)$.

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Proposition 141   If $S$ is infinite, then there is a monomorphism $f:\ensuremath{\mathbb{N}}\to S$.

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Proposition 142   is infinite.

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Proposition 143   If $f:A\to B$ is onto and $B$ is infinite, then $A$ is infinite.

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Proposition 144   If $S$ is infinite then there is a strictly decreasing sequence of subsets of $S$:

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

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Proposition 145   If $S$ is infinite then there is a strictly increasing sequence of subsets of $S$:

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

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Proposition 146   If $A\cup B$ is infinite, then either $A$ is infinite or $B$ is infinite.

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