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