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
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in which ω is the smallest infinite ordinal. Any solution to this equation has Cantor normal form
The least such ordinal is ε0 (pronounced epsilon nought or epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals:
Larger ordinal fixed points of the exponential map are indexed by ordinal subscripts, resulting in
The smallest epsilon number ε0 appears 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).
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 additively indecomposable ordinal#Multiplicatively indecomposable) 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. His gamma numbers are those of the form ωβ, and his delta numbers are those of the form ωωβ.
Ordinal ε numbers
The standard definition of ordinal exponentiation with base α is:
From this definition, it follows that for any fixed ordinal α > 1, the mapping
in which every element is the image of its predecessor under the mapping
The next epsilon number after
in which the sequence is again constructed by repeated base ω exponentiation but starts at
A different sequence with the same supremum,
The epsilon number
An epsilon number indexed by a limit ordinal α is constructed differently. The number
Since the epsilon numbers are an unbounded subclass of the ordinal numbers, they are enumerated using the ordinal numbers themselves. For any ordinal number
The following facts about epsilon numbers are very straightforward to prove:
Representation by trees
Using Cantor normal forms, the ordinals less than ε0 can be represented by finite rooted trees. The ordinal ωα + ωβ + ... + ωμ with α≥β≥...≥μ is represented by the tree whose root joins the trees of the ordinals α, β, ... μ at their roots.
Veblen hierarchy
The fixed points of the "epsilon mapping"
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). The least ordinal not reachable from 0 by this procedure—i. e., the least ordinal α for which φα(0)=α, or equivalently the first fixed point of the map
Surreal ε 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. One such function is the
It is natural to consider any fixed point of this expanded map to be an epsilon number, whether or not it happens to be strictly an ordinal number. Some examples of non-ordinal epsilon numbers are
and
There is a natural way to define