Formally, the definition is by transfinite induction : the \ gamma-th element of the class is defined ( provided it has already been defined for all \ beta ), as the smallest element greater than the \ beta-th element for all \ beta.
42.
Strictly speaking, it is not necessary in transfinite induction to prove the basis, because it is a vacuous special case of the proposition that if " P " is true of all, then " P " is true of " m ".
43.
The ordinal ? 0 is important for various reasons in arithmetic ( essentially because it measures the proof-theoretic strength of the Peano arithmetic : that is, Peano's axioms can show transfinite induction up to any ordinal less than ? 0 but not up to ? 0 itself ).
44.
In practice, the importance of well-ordering is justified by the possibility of applying transfinite induction, which says, essentially, that any property that passes on from the predecessors of an element to that element itself must be true of all elements ( of the given well-ordered set ).
45.
The method used in this proof can also be used to prove a cut elimination result for Peano arithmetic in a stronger logic than first-order logic, but the consistency proof itself can be carried out in ordinary first-order logic using the axioms of primitive recursive arithmetic and a transfinite induction principle.
46.
By transfinite induction, for each ordinal ? compute the set of positions where player 1 can force a win in ? steps, where a position with player 2 to move is losing ( for player 2 ) in ? steps iff for every move the resulting position is losing in less than ? steps.
47.
Certain computable ordinals are so large that while they can be given by a certain ordinal notation " o ", a given formal system might not be sufficiently powerful to show that " o " is, indeed, an ordinal notation : the system does not show transfinite induction for such large ordinals.
48.
For example, the consistency of the Peano arithmetic can be proved in Zermelo Fraenkel set theory ( ZFC ), or in theories of arithmetic augmented with transfinite induction, as in Gentzen's consistency proof . " from the article about G�dels theorems . talk ) 06 : 52, 7 May 2014 ( UTC ) Thomas Limberg ( Schmogrow)
49.
Whether the consistency of arithmetic is settled is of course subject to debate, but Gentzen's consistency proof ( using what amounts to structural induction on formulas, don't flip out at the term " transfinite induction " since there are no completed infinities involved ) and G�del's talk ) 01 : 15, 18 February 2010 ( UTC)
50.
Another class of examples of my point is the way that algebra classes, following Bourbaki, bundle all uses of the axiom of choice into the clunky and intuitively obscure Zorn's lemma, applications of which can almost always be replaced by a simple and clear transfinite induction in which you make a choice at each step .-- talk ) 19 : 12, 4 October 2008 ( UTC)
