Epsilon nought
From Wikipedia, the free encyclopedia
- This article is about an ordinal in mathematics. For the physics constant ε0, see permittivity.
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
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 (as in Gentzen's consistency proof and the proof of Goodstein's theorem). Its use by Gentzen to prove the consistency of Peano Arithmetic, along with Gödel's second incompleteness theorem, show that Peano Arithmetic cannot prove the well-foundedness of this ordering (it is in fact the least ordinal with this property, and as such, in proof-theoretic ordinal analysis, is used as a measure of the strength of the theory of Peano Arithmetic).
This was created by the German 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.
[edit] See also
[edit] 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.
