Search results
Results from the WOW.Com Content Network
Transfinite induction requires proving a base case (used for 0), a successor case (used for those ordinals which have a predecessor), and a limit case (used for ordinals which don't have a predecessor). Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.
In mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals , which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals , which are ordinal numbers used to provide an ordering ...
This form of induction, when applied to a set of ordinal numbers (which form a well-ordered and hence well-founded class), is called transfinite induction. It is an important proof technique in set theory, topology and other fields. Proofs by transfinite induction typically distinguish three cases:
In set theory, -induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property. Considered as an axiomatic principle, it is called the axiom schema of set induction. The principle implies transfinite induction and recursion.
Transfinite induction, an extension of mathematical induction to well-ordered sets Transfinite recursion; Transfinite arithmetic, the generalization of elementary arithmetic to infinite quantities; Transfinite interpolation, a method in numerical analysis to construct functions over a planar domain so that they match a given function on the ...
Get answers to your AOL Mail, login, Desktop Gold, AOL app, password and subscription questions. Find the support options to contact customer care by email, chat, or phone number.
The smallest epsilon number ε 0 appears in many induction proofs, because for many purposes transfinite induction is only required up to ε 0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem).
© 2012 Catalyst Inc. – Do Not Distribute Without Permission 2 Why Diversity Matters ..... 3