Definitions

Χ² Capricorni

Ε₀

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 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

varepsilon = omega^varepsilon,
in which ω is the smallest transfinite ordinal. Any solution to this equation has Cantor normal form varepsilon = omega^{varepsilon}.

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:

varepsilon_0 = omega^{omega^{omega^cdots}} = sup { omega, omega^{omega}, omega^{omega^{omega}}, omega^{omega^{omega^omega}}, dots }

Larger ordinal fixed points of the exponential map are indexed by ordinal subscripts, resulting in ε1, ε2, ... εω, εω+1, ... varepsilon_{varepsilon_0}, .... The ordinal ε0 is still countable, as is any epsilon number whose index is countable (there exist uncountable ordinals, and uncountable epsilon numbers whose index is an uncountable ordinal).

The smallest epsilon number ε0 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).

A more general class of epsilon numbers has been identified by John Horton Conway in the surreal number system, consisting of all surreals that are fixed points of the base ω exponential map x → ωx.

Ordinal ε numbers

The standard definition of ordinal exponentiation with base α>1 is:

  • alpha^0 = 1 ,,
  • alpha^{beta+1} = alpha^beta cdot alpha,
  • alpha^kappa = sup_{lambda < kappa} alpha^lambda for limit kappa.

From this definition, it follows that for any fixed ordinal α, the mapping beta mapsto alpha^beta is a normal function, so it has arbitrarily large fixed points by the fixed-point lemma for normal functions. When alpha = omega, these fixed points are precisely the ordinal epsilon numbers. The smallest of these, ε₀, is the supremum of the sequence

0, omega^0 = 1, omega^1 = omega, omega^omega, omega^{omega^omega}, ldots, omega uparrow uparrow k, ldots

in which every element is the image of its predecessor under the mapping beta mapsto omega^beta. (The general term is given using Knuth's up-arrow notation; the uparrow uparrow operator is equivalent to tetration.) Just as ωω is defined as the supremum of { ωk } for natural numbers k, the smallest ordinal epsilon number ε₀ may also be denoted omega uparrow uparrow omega; this notation is much less common than ε₀.

The next epsilon number after varepsilon_0 is

varepsilon_1 = sup{varepsilon_0 + 1, varepsilon_0 cdot omega, {varepsilon_0}^omega, {varepsilon_0}^{{varepsilon_0}^omega}, ldots},
in which the sequence is again constructed by repeated base ω exponentiation but starts at varepsilon_0 + 1 instead of at 0. A different sequence with the same supremum is obtained by starting from 0 and exponentiating with base ε₀ instead:
varepsilon_1 = sup{0, 1, varepsilon_0, {varepsilon_0}^{varepsilon_0}, {varepsilon_0}^{{varepsilon_0}^{varepsilon_0}}, ldots},
The epsilon number varepsilon_{alpha + 1} indexed by any successor ordinal α+1 is constructed similarly, by base ω exponentiation starting from varepsilon_alpha + 1 (or by base varepsilon_alpha exponentiation starting from 0).

An epsilon number indexed by a limit ordinal α is constructed differently. The number varepsilon_alpha is the supremum of the set of epsilon numbers { varepsilon_beta, beta < alpha }. The first such number is varepsilon_omega; its form as a tetration expression appears below. Whether or not the index α is a limit ordinal, varepsilon_alpha is a fixed point not only of base ω exponentiation but also of base γ exponentiation for all ordinals 1 < gamma < varepsilon_alpha.

Since the epsilon numbers are an unbounded subclass of the ordinal numbers, they are enumerated using the ordinal numbers themselves. For any ordinal number alpha, varepsilon_alpha is the least epsilon number (fixed point of the exponential map) not already in the set { varepsilon_beta, beta < alpha }. 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 very straightforward to prove:

  • Although it is quite a large number, varepsilon_0 is still countable, being a countable union of countable ordinals; in fact, varepsilon_alpha is countable if and only if alpha is countable.
  • The union (or supremum) of any nonempty set of epsilon numbers is an epsilon number; so for instance

varepsilon_omega = sup{varepsilon_0, varepsilon_1, varepsilon_2, ldots}
is an epsilon number. Thus, the mapping n mapsto varepsilon_n is a normal function.

The construction using the base ω exponential map is the original, due to Cantor; but one can obtain additional perspective using the base 2 exponential map. Its fixed points consist of ω itself, followed by the epsilon numbers:

begin{align}
omega = omega uparrow uparrow 1 & = sup { (0, 1, 2,) , 4, 16, ldots, 2 uparrow uparrow k , (k ge 2), ldots } varepsilon_0 = omega uparrow uparrow omega & = sup { (omega + 1, omega cdot 2, omega^2,) , omega^omega, omega^{omega^omega}, ldots, omega uparrow uparrow k , (k ge 2), ldots } varepsilon_1 = omega uparrow uparrow omega^2 & = sup { (varepsilon_0 + 1, varepsilon_0 cdot 2, {varepsilon_0}^2,) , {varepsilon_0}^{varepsilon_0}, ldots, varepsilon_0 uparrow uparrow k , (k ge 2), ldots } end{align}

The elements in parentheses are part of the exponential sequence but do not fit the general term obtained by tetration. (This is to be expected, since x uparrow uparrow k is 1 when k=0 and x when k=1.)

An epsilon number indexed by a limit ordinal is the supremum not of an exponential sequence but of a sequence of epsilon numbers:

begin{align}
varepsilon_omega = omega uparrow uparrow omega^omega & = sup { omega, varepsilon_0, varepsilon_1, varepsilon_2, ldots, omega uparrow uparrow omega^k, ldots } varepsilon_{omega^omega} = omega uparrow uparrow omega^{omega^omega} & = sup { varepsilon_0, varepsilon_omega, varepsilon_{omega^2}, ldots, omega uparrow uparrow omega^{omega^k}, ldots } varepsilon_{varepsilon_0} = omega uparrow uparrow uparrow 3 = omega uparrow uparrow (omega uparrow uparrow omega) & = sup { varepsilon_0, varepsilon_omega, varepsilon_{omega^omega}, ldots, omega uparrow uparrow (omega uparrow uparrow k), ldots } end{align}

The general formula is thus

varepsilon_alpha = omega uparrow uparrow omega^{(1+alpha)},
in which the term (1+α) is calculated using the rules of ordinal arithmetic, in which (1+α)=α for any transfinite ordinal α. The "1+" therefore essentially compensates for the historical naming convention, in which omega uparrow uparrow omega^1 has been labeled ε₀ rather than ε1.

The fixed points of the "epsilon mapping" x mapsto varepsilon_x form a normal function, whose fixed points form a normal function, whose …; this is known as the Veblen hierarchy (the Veblen functions with base φ0(α)=ωα). In the notation of the Veblen hierarchy, the epsilon mapping is φ1, and its fixed points are enumerated by φ2. Observing that the epsilon mapping coincides, for x > varepsilon_0, with x rightarrow omega uparrow uparrow x, one can speak instead of the fixed points of this "base ω tetration" map. The smallest few fixed points of the epsilon mapping are:

phi_2(0) = omega uparrow uparrow uparrow omega,,
phi_2(1) = (omega uparrow uparrow uparrow omega) uparrow uparrow uparrow omega = omega uparrow uparrow uparrow omega^2,, ldots

The process of defining successively "stronger" normal functions φ0, φ1, φ2, ... (or equivalently x ,rightarrow, omega^x, x ,rightarrow, omega uparrow uparrow x, x ,rightarrow, omega uparrow uparrow uparrow x, ..., which are not quite the same maps for small x but have the same fixed points) leads to a sequence of least fixed points:

{phi_1(0) = epsilon_0 = omega uparrow uparrow omega,, phi_2(0) = omega uparrow uparrow uparrow omega,, phi_3(0) = omega uparrow^4 omega,, ldots}.
The supremum of this sequence, in a set theory which permits transfinite recursion on such a sequence, is a rather larger ordinal which one might denote by phi_omega(0) = omega uparrow^omega omega; this ordinal is the smallest number that is a fixed point of x ,rightarrow, omega uparrow^k x for all natural numbers k.

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 alpha ,rightarrow, phi_alpha(0)—is the Feferman–Schütte ordinal Γ0. In a set theory where such an ordinal can be proven to exist, one has a map Γ that enumerates the fixed points Γ0, Γ1, Γ2, ... of alpha ,rightarrow, phi_alpha(0); 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.

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 the omega-map n mapsto omega^n; 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.

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

varepsilon_{-1} = {0, 1, omega, omega^omega, ldots mid varepsilon_0 - 1, omega^{varepsilon_0 - 1}, ldots}
and
varepsilon_{frac{1}{2}} = {varepsilon_0 + 1, omega^{varepsilon_0 + 1}, ldots mid varepsilon_1 - 1, omega^{varepsilon_1 - 1}, ldots}.

There is a natural way to define varepsilon_n for 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.

The surreal epsilon numbers form a subclass of the class of surreals that are fixed points of the base 2 exponential map x → 2x (or equivalently of the base y exponential map for any other finite base y > 1). This latter class is the most general idea of "epsilon number" (exponential fixed point), in the same sense in which the surreals are the most general kind of "number" (ordered field); but it differs from the ordinal epsilon numbers in critical respects (beginning with the fact that ω is a fixed point of x → 2x but not of x → ωx). The fixed points of x → 2x can also be indexed by the surreals; for every surreal x with birthday α, there is a fixed point F(x) with birthday omega uparrow uparrow omega^alpha.

If the birthday of x is a successor ordinal α+1, Sα contains either a greatest element y such that y<x or a least element z such that z>x (or both). The fixed point F(x) is the exponential attractor of F(y)+1 and/or F(z)−1, i. e., the surreal number obtained by transfinite recursion from a series formed by iterated base 2 exponentiation starting at F(y)+1 or F(z)−1. (More precisely, iterated exponentiation attracts any non-fixed-point surreal number to one of the gaps above the reals in the surreal number system. The exponential attractor is the result of transfinite recursion "into the gap", i. e., the simplest surreal number lying beyond the gap.) As usual, the case where the birthday of x is a limit ordinal α is handled using Conway cuts in the transfinite union of previous generations.

The unique fixed point of x → 2x with birthday omega uparrow uparrow omega^0 = omega is ω itself, which we may label F(0). The two fixed points with birthday omega uparrow uparrow omega^1 = varepsilon_0 are omega^{omega^{varepsilon_0}} = varepsilon_0 and omega^{omega^{-varepsilon_0}} (the smallest infinite number in generation ε₀); they are the exponential attractors of ω+1 and ω−1 respectively, and may be labeled F(1) and F(−1). The largest and smallest infinite numbers in generation ε1 are F(2)=ε1 and F(−2) respectively; Conway's ε−1 also appears in generation ε1 as F(1/2), and his varepsilon_{frac{1}{2}} appears in generation ε2 as F(3/2).

See also

References

  • J.H. Conway, On Numbers and Games (1976) Academic Press ISBN 0-12-186350-6

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