Finite

Definition 63   A set $S$ is finite if for every $f:S\to S$ and every $s\in S$ the sequence defined inductively by $s_0=s, s_{n+1}=f(s_n)$ is eventually periodic; that is, there are natural numbers $k$ and $p\neq 0$ such that $s_k=s_{p+k}$.

This is taking as a definition the observation that infinite cycles and chaos are not possible in finite dynamical systems. Definition of what it means for a set to be finite is (surprizingly) much more complicated than definition of infinite. This particular definition is not one of the standard ones from the literature (which usually uses isomorphism with finite cardinals, though numerous other definitions are also common).

Proposition 147   The empty set $\emptyset$ is finite.

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Theorem 148   If a set $S$ is infinite, then it is not finite.

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Definition 64   The the cardinal set $[n]=\{k\in \ensuremath{\mathbb{N}}\vert k<n\}$. Note that $[n]$ has exactly $n$ elements.

Proposition 149   The cardinal $[n]$ is finite.

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Proposition 150   If $A$ is finite and $B\subseteq A$ then $B$ is finite.

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Proposition 151   If $A$ is finite and $f:A\to B$ is onto, then $B$ is finite.

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Proposition 152   If $A$ and $B$ are finite then so are $A\times B$, $A+B$, and $A\cup B$.

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Theorem 153   A set $S$ is finite only if it is not infinite.

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Hint: This one is hard. It will require use of the axiom of choice. $\spadesuit$