| 1. | There are models of ZF in which the axiom of choice fails.
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| 2. | If the axiom of choice holds, then every successor cardinal is regular.
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| 3. | Without the axiom of choice, there are cardinals which cannot be well-ordered.
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| 4. | The construction of " f " relies on the axiom of choice.
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| 5. | However, these may all be different if the axiom of choice fails.
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| 6. | Indeed, their existence is a non-trivial consequence of the axiom of choice.
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| 7. | The Axiom of Choice can be proven from the well-ordering theorem as follows.
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| 8. | One consequence of Shoenfield's theorem relates to the axiom of choice.
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| 9. | Meanwhile, the negation of the axiom of choice is, curiously, an NF theorem.
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| 10. | But none of these theorems actually proves that the axiom of choice holds.
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