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DanielLC comments on You only need faith in two things - Less Wrong Discussion
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Comments (86)
How?
You can do it with the axiom of choice, but beyond that I'm pretty sure you can't.
If "arbitrary size" means "arbitrarily large size," see Hartogs numbers. On the other hand, the well-ordering principle is equivalent to AC.
Take the empty set. Add an element. Preserving the order of existing elements, add a greatest element. Repeat.
That sounds like it would only work for countable sets.
Is the single large ordinal which must be well-ordered uncountable? I had figured that simply unbounded was good enough for this application.