| 31. | The continuum hypothesis and the axiom of choice, are examples of possible transcendental decision points.
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| 32. | This has been used as an argument against the use of the axiom of choice.
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| 33. | A form of the axiom of choice is a theorem, yet excluded middle is not.
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| 34. | A parallel example uses the axiom of choice ( AC ) and ZF set theory.
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| 35. | Two famous statements in set theory are the axiom of choice and the continuum hypothesis.
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| 36. | In ZF without the axiom of choice, it is possible that every ultrafilter is principal.
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| 37. | The proof of this requires the axiom of choice.
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| 38. | The axiom of choice asserts the existence of such elements; it is therefore equivalent to:
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| 39. | One of those proofs relies on ultraproducts hinging on the axiom of choice as follows:
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| 40. | The properties of total boundedness mentioned above rely in part on the axiom of choice.
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