The statement that every set can be well-ordered is called the well-ordering principle. It is not a theorem, but an axiom of set theory equivalent to the axiom of choice.
Any countable set can be well-ordered by mapping it to the set of natural numbers, so the principle becomes more interesting when it is applied to uncountable sets, such as the set of real numbers, because even when the principle is assumed it can be shown that there is no formula for expressing such an order.
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