Epsilon number
In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map.Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication.The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ε that satisfy the equation in which ω is the smallest infinite ordinal.The least such ordinal is ε0 (pronounced epsilon nought (chiefly British), epsilon naught (chiefly American), or epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals: where sup is the supremum, which is equivalent to set union in the case of the von Neumann representation of ordinals.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).Many larger epsilon numbers can be defined using the Veblen function.A more general class of epsilon numbers has been identified by John Horton Conway and Donald Knuth in the surreal number system, consisting of all surreals that are fixed points of the base ω exponential map x → ωx.Hessenberg (1906) defined gamma numbers (see additively indecomposable ordinal) to be numbers γ > 0 such that α + γ = γ whenever α < γ, and delta numbers (see multiplicatively indecomposable ordinal) to be numbers δ > 1 such that αδ = δ whenever 0 < α < δ, and epsilon numbers to be numbers ε > 2 such that αε = ε whenever 1 < α < ε.The standard definition of ordinal exponentiation with base α is: From this definition, it follows that for any fixed ordinal α > 1, the mappingis a normal function, so it has arbitrarily large fixed points by the fixed-point lemma for normal functions., these fixed points are precisely the ordinal epsilon numbers., is obtained by starting from 0 and exponentiating with base ε0 instead: Generally, the epsilon numberis the least epsilon number (fixed point of the exponential map) not already in the setIt might appear that this is the non-constructive equivalent of the constructive definition using iterated exponentiation; but the two definitions are equally non-constructive at steps indexed by limit ordinals, which represent transfinite recursion of a higher order than taking the supremum of an exponential series.The following facts about epsilon numbers are straightforward to prove: Any epsilon number ε has Cantor normal form, which means that the Cantor normal form is not very useful for epsilon numbers.The ordinals less than ε0, however, can be usefully described by their Cantor normal forms, which leads to a representation of ε0 as the ordered set of all finite rooted trees, as follows.in turn has a similar Cantor normal form.We obtain the finite rooted tree representing α by joining the roots of the trees representingis represented by a tree containing a root and a single leaf.)This representation is related to the proof of the hydra theorem, which represents decreasing sequences of ordinals as a graph-theoretic game.In the notation of the Veblen hierarchy, the epsilon mapping is φ1, and its fixed points are enumerated by φ2.Continuing in this vein, one can define maps φα for progressively larger ordinals α (including, by this rarefied form of transfinite recursion, limit ordinals), with progressively larger least fixed points φα+1(0).In a set theory where such an ordinal can be proved to exist, one has a map Γ that enumerates the fixed points Γ0, Γ1, Γ2, ... of; these are all still epsilon numbers, as they lie in the image of φβ for every β ≤ Γ0, including of the map φ1 that enumerates epsilon numbers.In On Numbers and Games, the classic exposition on surreal numbers, John Horton Conway provided a number of examples of concepts that had natural extensions from the ordinals to the surreals.; this mapping generalises naturally to include all surreal numbers in its domain, which in turn provides a natural generalisation of the Cantor normal form for surreal numbers.Some examples of non-ordinal epsilon numbers are and There is a natural way to definefor every surreal number n, and the map remains order-preserving.Conway goes on to define a broader class of "irreducible" surreal numbers that includes the epsilon numbers as a particularly interesting subclass.