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The last two steps can be reformulated as one step:
This is in fact the most general form of mathematical induction and it can be shown that it is not only valid for statements about natural numbers, but for statements about elements of any well-founded set, that is, a set with a partial order that contains no infinite descending chains (where < is defined such that a < b iff a ≤ b and a ≠ b).
This form of induction, when applied to ordinals (which form a well-ordered and hence well-founded class), is called transfinite inductionSet theory Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. It may be regarded as one of three forms of m. It is an important proof technique in set theorySet theory is the mathematical theory of sets, which represent collections of abstract objects. It has a central role in modern mathematical theory, providing the basic language in which most of mathematics is expressed. For more information on set theory, topologyTopology is the study or science of places. It derives its name from the Greek words τοπος meaning place and λογος meaning study, talk. See also earth science, geography, human geography, g and other fields.
Proofs by transfinite induction typically need to distinguish three cases:
See also three forms of mathematical inductionProofs that a subset of { 1, 2, 3,. is in fact the whole set { 1, 2, 3,. by mathematical induction usually have one of the following three forms. The basis for induction is trivial; the substantial part of the proof goes from case n to case n + 1. The cas.
The principle of mathematical induction is usually stated as an axiomFor the algebra software named Axiom, see Axiom (algebra software). For the 1970s Australian rock music group, see Axiom (band). In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up of natural numbers, see Peano axiomsIn mathematics, the Peano axioms (or Peano postulates are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic . This theory constitutes a fundamental formalism for ari. However, it can be proved in some logical systems; for instance, if the following axiom:
is employed.
Note that the additional axiom is indeed an alternative formulation of principle of mathematical induction. That is, the two are equivalent. See proof of mathematical inductionThe principle of mathematical induction can be proved if the following axiom is assumed: :The set of all natural numbers is well-ordered (that is, every non-empty set of natural numbers has a least element). A simplified version is given here. This proof.
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