| 1. | This may require a separate transfinite induction argument to justify rigorously.
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| 2. | Then transfinite induction tells us that P is true for all ordinals.
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| 3. | But there's no excuse not to be comfortable with transfinite induction!
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| 4. | The uniqueness of the sequence satisfying these properties can be proved using transfinite induction.
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| 5. | This distinction is important, because many definitions by transfinite induction rely upon it.
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| 6. | This is important because it makes every set susceptible to the powerful technique of transfinite induction.
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| 7. | We create a counterexample by transfinite induction on the set of strategies under this well ordering:
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| 8. | Transfinite induction can be used not only to prove things, but also to define them.
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| 9. | He then proves by transfinite induction on these ordinals that no proof can conclude in a contradiction.
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| 10. | This group of definitions is dyadic fractions; a wider universe is reachable given some form of transfinite induction.
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