Well-Ordering Principle

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The well-ordering principle states that every set is well-ordered by some relation. It is an equivalent of the axiom of choice.

Example

For example, the well-ordering principle states that $\mathbb R$, the set of reals, is well-ordered. This can be done in a very nonconstructive way:

Let $f:\mathbb R\rightarrow\beth_1$ be a bijection. Then, define $R=\{(a,b):f(a)<f(b)\}$. Because $f$ preserves order between $(\mathbb R,R)$ and $(\beth_1,<)$ by definition of $R$, then $R$ must be a well-ordering of $\mathbb R$.

The instance of the axiom of choice here is in the choice of $f$. (Elaborate?)


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