| 11. | Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers.
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| 12. | The uniqueness of the sequence satisfying these properties can be proved using transfinite induction.
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| 13. | This is again a definition by transfinite recursion.
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| 14. | Like " oh, you study infinite-dimensional hyperchaotic transfinite differential matrix supermanifolds?
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| 15. | This distinction is important, because many definitions by transfinite induction rely upon it.
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| 16. | In group theory, Hamkins proved that every group has a terminating transfinite automorphism tower.
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| 17. | This is important because it makes every set susceptible to the powerful technique of transfinite induction.
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| 18. | One may continue into the transfinite.
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| 19. | ? 1 1-CA 0 is stronger than arithmetical transfinite recursion and is fully impredicative.
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| 20. | We create a counterexample by transfinite induction on the set of strategies under this well ordering:
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