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More formally stated: "Suppose $A \subset C$ with $|A|=n$. Then one could identify each element in $A$ as $a_1,...,a_n$ such that $a_i for $1\leq i\leq n-1$."

Sorry if the $\LaTeX$ is not up to par, but I'm still a novice at typesetting.

Anyway, this is what I understand to be a way to state the well ordered theorem for a finite set through induction. I had the idea that one could try and examine each element and put them in order since the set isn't infinite, but it might not be the case. This may seem trivial, but I just have a difficulty wrapping my head around it. Help is appreciated.

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    @Vilid Yes I understand what you mean. I read about this here: http://www.proofwiki.org/wiki/Equivalence_of_Well-Ordering_Principle_and_Induction#Final_assembly still they prove induction is true previously, but for an infinite set.2012-09-17

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Let $\bar{n} = \{1, ..., n\}$. If you know that $\mathbb{N}$ is a well-ordering, then the ordering restricted to $\bar{n}$ is a well-ordering for each $n \in \mathbb{N}$. By the way, you can show that the well-ordering of $\mathbb{N}$ is equivalent to induction on $\mathbb{N}$.


If $A$ is a finite set, then there exists a bijection $f : A \rightarrow \bar{n}$ for some $n \in \mathbb{N}$.

Define the ordering $\prec$ on $A$ by $a \prec a'$ if and only if $f(a) <_\mathbb{N} f(a')$. Then $\prec$ is a well-ordering of $A$.