Epsilon numbers (mathematics)

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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

\varepsilon = \omega^\varepsilon, \,

in which ω is the smallest infinite 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^{\cdot^{\cdot^\cdot}}}} = \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 \varepsilon_1, \varepsilon_2,\ldots,\varepsilon_\omega, \varepsilon_{\omega+1}, \ldots, \varepsilon_{\varepsilon_0}, \ldots, \varepsilon_{\varepsilon_1}, \ldots, \varepsilon_{\varepsilon_{\varepsilon_{\cdot_{\cdot_{\cdot}}}}},\ldots. 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).

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.

Ordinal ε numbers[edit]

The standard definition of ordinal exponentiation with base α is:

From this definition, it follows that for any fixed ordinal α > 1, 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, \omega^{\varepsilon_0 + 1}, \omega^{\omega^{\varepsilon_0 + 1}}, \omega^{\omega^{\omega^{\varepsilon_0 + 1}}}, \dots\},

in which the sequence is again constructed by repeated base ω exponentiation but starts at \varepsilon_0 + 1 instead of at 0. Notice

\omega^{\varepsilon_0 + 1} = \omega^{\varepsilon_0} \cdot \omega^1 = \varepsilon_0 \cdot \omega \,,
\omega^{\omega^{\varepsilon_0 + 1}} = \omega^{(\varepsilon_0 \cdot \omega)} = {(\omega^{\varepsilon_0})}^\omega = \varepsilon_0^\omega \,,
\omega^{\omega^{\omega^{\varepsilon_0 + 1}}} = \omega^{{\varepsilon_0}^\omega} = \omega^{{\varepsilon_0}^{1+\omega}} = \omega^{(\varepsilon_0\cdot{\varepsilon_0}^\omega)} = {(\omega^{\varepsilon_0})}^{{\varepsilon_0}^\omega} = {\varepsilon_0}^{{\varepsilon_0}^\omega} \,.

A different sequence with the same supremum, \varepsilon_1, 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).

\varepsilon_{\alpha + 1} = \sup\{\varepsilon_\alpha + 1, \omega^{\varepsilon_\alpha + 1}, \omega^{\omega^{\varepsilon_\alpha + 1}}, \dots\} = \sup\{0, 1, \varepsilon_\alpha, \varepsilon_\alpha^{\varepsilon_\alpha}, \varepsilon_\alpha^{\varepsilon_\alpha^{\varepsilon_\alpha}}, \dots\}

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. 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.
1 \leq \alpha \rightarrow \epsilon_{\omega_{\alpha}} = \omega_{\alpha} \,.

Veblen hierarchy[edit]

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.

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[edit]

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 \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.

See also[edit]

References[edit]

  • J.H. Conway, On Numbers and Games (1976) Academic Press ISBN 0-12-186350-6
  • Section XIV.20 of Sierpiński, Wacław (1965), Cardinal and ordinal numbers (Second revised ed.), PWN – Polish Scientific Publishers 

External links[edit]