transfinite induction การใช้

• This may require a separate transfinite induction argument to justify rigorously.
• Then transfinite induction tells us that P is true for all ordinals.
• But there's no excuse not to be comfortable with transfinite induction!
• The uniqueness of the sequence satisfying these properties can be proved using transfinite induction.
• This distinction is important, because many definitions by transfinite induction rely upon it.
• This is important because it makes every set susceptible to the powerful technique of transfinite induction.
• We create a counterexample by transfinite induction on the set of strategies under this well ordering:
• Transfinite induction can be used not only to prove things, but also to define them.
• He then proves by transfinite induction on these ordinals that no proof can conclude in a contradiction.
• This group of definitions is dyadic fractions; a wider universe is reachable given some form of transfinite induction.
• The only reason I can see to prefer it is if you're not comfortable with transfinite induction.
• The well-ordering of the values of single parameters is provided by the inductive hypothesis of the transfinite induction.
• One common use of the Borel hierarchy is to prove facts about the Borel sets using transfinite induction on rank.
• We proceed by transfinite induction on ? to show that ? is ?-inaccessible for any ? d " ?.
• Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction.
• Ordinal analysis was originated by Genzten, who proved the consistency of Peano Arithmetic using transfinite induction up to ordinal ? 0.
• It then follows by transfinite induction that there is one and only one function satisfying the recursion formula up to and including ?.
• Trying to create a general theory, Denjoy used transfinite induction over the possible types of singularities, which made the definition quite complicated.
• Such a definition is normally said to be by transfinite recursion  the proof that the result is well-defined uses transfinite induction.
• In 1936, Gerhard Gentzen gave a proof of the consistency of Peano's axioms, using transfinite induction up to an ordinal called ? 0.