:The original proposals of the formalists to make classical mathematics secure by a consistency proof did not contemplate that such a method as transfinite induction up to ? 0 would have to be used.
32.
It can be shown by transfinite induction that every well-ordered set is order-isomorphic to exactly one of these ordinals, that is, there is an order preserving bijective function between them.
33.
The presentations of the von Neumann universe by Bernays and Mendelson both give credit to von Neumann for the transfinite induction construction method, although not for its application to the construction of the universe of ordinary sets.
34.
Gentzen showed that it is possible to produce a proof of the consistency of arithmetic in a finitary system augmented with axioms of transfinite induction, and the techniques he developed to do so were seminal in proof theory.
35.
When you're doing a careful proof by transfinite induction, or a construction by transfinite recursion, ordinarily you should probably have three parts to the proof : stage 0, successor stages, and limit stages.
36.
I was then using Hausdorf's maximality theorem to set up a transfinite induction argument ( So, if we have a partially ordered set, then this will contain a totally ordered subset which is maximimal ).
37.
If a set is well ordered ( or even if it merely admits a well-founded relation ), the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set.
38.
The process involves defining, for each countable ordinal, via transfinite induction, a set by " throwing in " all possible countable unions and complements, and taking the union of all that over all of "'? 1 " '.
39.
The integrity of the von Neumann universe depends fundamentally on the integrity of the ordinal numbers, which act as the rank parameter in the construction, and the integrity of transfinite induction, by which both the ordinal numbers and the von Neumann universe are constructed.
40.
In classical reverse mathematics, " bar induction " ( BI ) denotes the related principle stating that if a relation " R " is a well-order, then we have the schema of transfinite induction over " R " for arbitrary formulas.
