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