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Inmathematics, theepsilon numbers are a collection oftransfinite numbers whose defining property is that they arefixed points of anexponential 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 byGeorg Cantor in the context ofordinal arithmetic; they are theordinal numbersε that satisfy theequation

${\displaystyle \epsilon ={\omega }^{\epsilon },}$

in which ω is the smallest infinite ordinal.

The least such ordinal isε₀ (pronouncedepsilon nought (chiefly British),epsilon naught (chiefly American), orepsilon zero), which can be viewed as the "limit" obtained bytransfinite recursion from a sequence of smaller limit ordinals:

${\displaystyle {\epsilon }_0={\omega }^{{\omega }^{{\omega }^{{\cdot }^{{\cdot }^{\cdot }}}}}=sup\left\{\omega ,{\omega }^{\omega },{\omega }^{{\omega }^{\omega }},{\omega }^{{\omega }^{{\omega }^{\omega }}},\dots \right\},}$

wheresup is thesupremum, which is equivalent toset union in the case of the von Neumann representation of ordinals.

Larger ordinal fixed points of the exponential map are indexed by ordinal subscripts, resulting in${\displaystyle {\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_{\omega },{\epsilon }_{\omega +1},\dots ,{\epsilon }_{{\epsilon }_0},\dots ,{\epsilon }_{{\epsilon }_1},\dots ,{\epsilon }_{{\epsilon }_{{\epsilon }_{{\cdot }_{{\cdot }_{\cdot }}}}},\dots {\zeta }_0={\phi }_2(0)}$.^[1] The ordinalε₀ is stillcountable, as is any epsilon number whose index is countable.Uncountable ordinals also exist, along with uncountable epsilon numbers whose index is an uncountable ordinal.

The smallest epsilon numberε₀ appears in manyinduction proofs, because for many purposestransfinite induction is only required up toε₀ (as inGentzen's consistency proof and the proof ofGoodstein's theorem). Its use byGentzen to prove the consistency ofPeano arithmetic, along withGödel's second incompleteness theorem, show that Peano arithmetic cannot prove thewell-foundedness of this ordering (it is in fact the least ordinal with this property, and as such, inproof-theoreticordinal analysis, is used as a measure of the strength of the theory of Peano arithmetic).

Many larger epsilon numbers can be defined using theVeblen function.

A more general class of epsilon numbers has been identified byJohn Horton Conway andDonald Knuth in thesurreal number system, consisting of all surreals that are fixed points of the base ω exponential mapx →ω^x.

Hessenberg (1906) defined gamma numbers (seeadditively indecomposable ordinal) to be numbersγ > 0 such thatα +γ =γ wheneverα <γ, and delta numbers (seemultiplicatively indecomposable ordinal) to be numbersδ > 1 such thatαδ =δ whenever0 <α <δ, and epsilon numbers to be numbersε > 2 such thatα^ε =ε whenever1 <α <ε. His gamma numbers are those of the formω^β, and his delta numbers are those of the formω^ωβ.

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The standard definition ofordinal exponentiation with base α is:

• ${\displaystyle {\alpha }^0=1,}$
• ${\displaystyle {\alpha }^{\beta }={\alpha }^{\beta -1}\cdot \alpha ,}$ when${\displaystyle \beta }$ has an immediate predecessor${\displaystyle \beta -1}$.
• , whenever${\displaystyle \beta }$ is alimit ordinal.

From this definition, it follows that for any fixed ordinalα > 1, themapping${\displaystyle \beta \mapsto {\alpha }^{\beta }}$ is anormal function, so it has arbitrarily largefixed points by thefixed-point lemma for normal functions. When${\displaystyle \alpha =\omega }$, these fixed points are precisely the ordinal epsilon numbers.

• ${\displaystyle {\epsilon }_0=sup\left\{1,\omega ,{\omega }^{\omega },{\omega }^{{\omega }^{\omega }},{\omega }^{{\omega }^{{\omega }^{\omega }}},\dots \right\},}$
• ${\displaystyle {\epsilon }_{\beta }=sup\left\{{\epsilon }_{\beta -1}+1,{\omega }^{{\epsilon }_{\beta -1}+1},{\omega }^{{\omega }^{{\epsilon }_{\beta -1}+1}},{\omega }^{{\omega }^{{\omega }^{{\epsilon }_{\beta -1}+1}}},\dots \right\},}$ when${\displaystyle \beta }$ has an immediate predecessor${\displaystyle \beta -1}$.
• , whenever${\displaystyle \beta }$ is a limit ordinal.

Because

${\displaystyle {\omega }^{{\epsilon }_0+1}={\omega }^{{\epsilon }_0}\cdot {\omega }^1={\epsilon }_0\cdot \omega ,}$

${\displaystyle {\omega }^{{\omega }^{{\epsilon }_0+1}}={\omega }^{({\epsilon }_0\cdot \omega )}={({\omega }^{{\epsilon }_0})}^{\omega }={\epsilon }_0^{\omega },}$

${\displaystyle {\omega }^{{\omega }^{{\omega }^{{\epsilon }_0+1}}}={\omega }^{{{\epsilon }_0}^{\omega }}={\omega }^{{{\epsilon }_0}^{1+\omega }}={\omega }^{({\epsilon }_0\cdot {{\epsilon }_0}^{\omega })}={({\omega }^{{\epsilon }_0})}^{{{\epsilon }_0}^{\omega }}={{\epsilon }_0}^{{{\epsilon }_0}^{\omega }},}$

a different sequence with the same supremum,${\displaystyle {\epsilon }_1}$, is obtained by starting from 0 and exponentiating with baseε₀ instead:

${\displaystyle {\epsilon }_1=sup\left\{1,{\epsilon }_0,{{\epsilon }_0}^{{\epsilon }_0},{{\epsilon }_0}^{{{\epsilon }_0}^{{\epsilon }_0}},\dots \right\}.}$

Generally, the epsilon number${\displaystyle {\epsilon }_{\beta }}$ indexed by any ordinal that has an immediate predecessor${\displaystyle \beta -1}$ can be constructed similarly.

${\displaystyle {\epsilon }_{\beta }=sup\left\{1,{\epsilon }_{\beta -1},{\epsilon }_{\beta -1}^{{\epsilon }_{\beta -1}},{\epsilon }_{\beta -1}^{{\epsilon }_{\beta -1}^{{\epsilon }_{\beta -1}}},\dots \right\}.}$

In particular, whether or not the index β is a limit ordinal,${\displaystyle {\epsilon }_{\beta }}$ is a fixed point not only of base ω exponentiation but also of base δ exponentiation for all ordinals.

Since the epsilon numbers are an unbounded subclass of the ordinal numbers, they are enumerated using the ordinal numbers themselves. For any ordinal number${\displaystyle \beta }$,${\displaystyle {\epsilon }_{\beta }}$ is the least epsilon number (fixed point of the exponential map) not already in the set. It 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:

• Although it is quite a large number,${\displaystyle {\epsilon }_0}$ is stillcountable, being a countable union of countable ordinals; in fact,${\displaystyle {\epsilon }_{\beta }}$ is countable if and only if${\displaystyle \beta }$ is countable.
• The union (or supremum) of anynon-empty set of epsilon numbers is an epsilon number; so for instance$${\displaystyle {\epsilon }_{\omega }=sup\{{\epsilon }_0,{\epsilon }_1,{\epsilon }_2,\dots \}}$$ is an epsilon number. Thus, the mapping${\displaystyle \beta \mapsto {\epsilon }_{\beta }}$ is a normal function.
• Theinitial ordinal of anyuncountablecardinal is an epsilon number.$${\displaystyle \alpha \ge 1\Rightarrow {\epsilon }_{{\omega }_{\alpha }}={\omega }_{\alpha }.}$$

Any epsilon number ε hasCantor normal form${\displaystyle \epsilon ={\omega }^{\epsilon }}$, which means that the Cantor normal form is not very useful for epsilon numbers. The ordinals less thanε₀, however, can be usefully described by their Cantor normal forms, which leads to a representation ofε₀ as the ordered set of allfinite rooted trees, as follows. Any ordinal has Cantor normal form${\displaystyle \alpha ={\omega }^{{\beta }_1}+{\omega }^{{\beta }_2}+\cdots +{\omega }^{{\beta }_k}}$ wherek is anatural number and${\displaystyle {\beta }_1,\dots ,{\beta }_k}$ are ordinals with${\displaystyle \alpha >{\beta }_1\ge \cdots \ge {\beta }_k}$, uniquely determined by${\displaystyle \alpha }$. Each of the ordinals${\displaystyle {\beta }_1,\dots ,{\beta }_k}$ in turn has a similar Cantor normal form. We obtain the finite rooted tree representing α by joining the roots of the trees representing${\displaystyle {\beta }_1,\dots ,{\beta }_k}$ to a new root. (This has the consequence that the number 0 is represented by a single root while the number${\displaystyle 1={\omega }^0}$ is represented by a tree containing a root and a single leaf.) An order on the set of finite rooted trees is defined recursively: we first order the subtrees joined to the root in decreasing order, and then uselexicographic order on these ordered sequences of subtrees. In this way the set of all finite rooted trees becomes awell-ordered set which isorder isomorphic toε₀.

This representation is related to the proof of thehydra theorem, which represents decreasing sequences of ordinals as agraph-theoretic game.

The fixed points of the "epsilon mapping"${\displaystyle x\mapsto {\epsilon }_x}$ form a normal function, whose fixed points form a normal function; this is known as theVeblen hierarchy (the Veblen functions with baseφ₀(α) =ω^α). In the notation of the Veblen hierarchy, the epsilon mapping isφ₁, and its fixed points are enumerated byφ₂ (seeordinal collapsing function.)

Continuing in this vein, one can define mapsφ_α for progressively larger ordinals α (including, by this rarefied form of transfinite recursion, limit ordinals), with progressively larger least fixed pointsφ_α+1(0). The least ordinal not reachable from 0 by this procedure—i. e., the least ordinal α for whichφ_α(0) =α, or equivalently the first fixed point of the map${\displaystyle \alpha \mapsto {\phi }_{\alpha }(0)}$—is theFeferman–Schütte ordinalΓ₀. In a set theory where such an ordinal can be proved to exist, one has a mapΓ that enumerates the fixed pointsΓ₀,Γ₁,Γ₂, ... of${\displaystyle \alpha \mapsto {\phi }_{\alpha }(0)}$; these are all still epsilon numbers, as they lie in the image ofφ_β for everyβ ≤ Γ₀, including of the mapφ₁ that enumerates epsilon numbers.

InOn Numbers and Games, the classic exposition onsurreal 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${\displaystyle \omega }$-map${\displaystyle n\mapsto {\omega }^n}$; this mapping generalises naturally to include all surreal numbers in itsdomain, which in turn provides a natural generalisation of theCantor normal form for surreal numbers.

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

${\displaystyle {\epsilon }_{-1}=\left\{0,1,\omega ,{\omega }^{\omega },\dots \mid {\epsilon }_0-1,{\omega }^{{\epsilon }_0-1},\dots \right\}}$

and

${\displaystyle {\epsilon }_{1/2}=\left\{{\epsilon }_0+1,{\omega }^{{\epsilon }_0+1},\dots \mid {\epsilon }_1-1,{\omega }^{{\epsilon }_1-1},\dots \right\}.}$

There is a natural way to define${\displaystyle {\epsilon }_n}$ for every surreal numbern, and the map remainsorder-preserving. Conway goes on to define a broader class of "irreducible" surreal numbers that includes the epsilon numbers as a particularly interesting subclass.

• Ordinal arithmetic
• Large countable ordinal

• Ordinal numbers

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