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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

This result could not be intuitively simpler, yet it depends on the axiom of choice for its truth!

Let

be the set of all equivalence classes under the relation ~ on(More precisely, it is the axiom of choice that guarantees the existence of an injective function

, from which we set .)Offline

**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

Zorn's lemma is obviously true, the Well-ordering principle is obviously false, and who knows about the Axiom of Choice.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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