In mathematics, ε₀ is the smallest transfinite ordinal number which cannot be reached from ω (the smallest transfinite ordinal) with a finite number of the ordinal operations of addition, multiplication and exponentiation. As such it is a limit ordinal. It is given by Euclid, detail from The School of Athens by Raphael. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... A limit ordinal is an ordinal number which is not a successor ordinal. ...

or in Cantor normal form by

The ordinal ε₀ is still countable (there exist uncountable ordinals). This ordinal is very important in many induction proofs, because for many purposes, transfinite induction is only required up to ε₀. In mathematics the term countable set is used to describe the size of a set, e. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. ... 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. ...

This was created by the Russian-born mathematician Georg Cantor. It is also frequently cited by the Argentine-American mathematician and computer scientist Gregory Chaitin in his lectures and papers. This ordinal is also called "epsilon zero". Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... Gregory J. Chaitin (born 1947) is an Argentine-American mathematician and computer scientist. ...

See also

• ordinal arithmetic

In the mathematical field of set theory, there are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. ...

External links

• "A Century of Controversy over the Foundations of Mathematics" — A lecture given by Gregory Chaitin on April 30, 1999 at University of Massachusetts at Lowell.