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It may not be obvious that it can be proven, without using AC, that there even exists a nonzero ordinal onto which there is no surjection from the reals (if there is such an ordinal, then there must be a least one because the ordinals are wellordered). However, suppose there were no such ordinal.
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of all ordinals into the set of all sets of orderings on the reals (which can to be seen to be a set via repeated application of the
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has a strong partition cardinal above it. This does not preclude the possibility that a single strong partition cardinal, above
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139:. The axiom of determinacy is equivalent to the existence of unboundedly many strong partition cardinals below
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Kechris, Alexander S.; Woodin, W. Hugh (2008), "The equivalence of partition properties and determinacy",
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shows that the class of all ordinals is in fact a set. But that is impossible, by the
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of all prewellorderings of the reals having order type α. This would give an
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Cunningham, Daniel W. (2017), "A strong partition cardinal above
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Games, scales, and Suslin cardinals: The Cabal
Seminar, Vol. I
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413:{\displaystyle \varTheta }
328:{\displaystyle \varTheta }
239:{\displaystyle \varTheta }
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172:{\displaystyle \varTheta }
152:{\displaystyle \varTheta }
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124:{\displaystyle \varTheta }
47:{\displaystyle \varTheta }
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74:{\displaystyle \alpha }
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