The hyperserial field of surreal numbers

Vincent BagayokoA, UMons, LIX

Joris van der HoevenB, CNRS, LIX



. This article has been written using GNU TeXmacs [27].

For any ordinal , we show how to define a hyperexponential and a hyperlogarithm on the class of positive infinitely large surreal numbers. Such functions are archetypes of extremely fast and slowly growing functions at infinity. We also show that the surreal numbers form a so-called hyperserial field for our definition.


The ordered field of surreal numbers was introduced by Conway in [11]. Conway originally used transfinite recursion to define both the surreal numbers (henceforth called numbers), the ordering on No, and the ring operations. For any two sets and of numbers with (i.e. for all and ), there exists a number with

and all numbers can be obtained in this way. Given and , we have

and similar recursive formulas exist for , and for deciding whether , , and . It is truly remarkable that turns out to be a totally ordered real-closed field for such “simple” definitions [11]. The bracket is called the Conway bracket. Using this bracket, we obtain a surreal number in any traditional Dedekind cut, which allows us to embed into . In addition, contains all ordinal numbers

so is actually a proper class.

An interesting question is which other operations from calculus can be extended to the surreal numbers. Gonshor has shown how to extend the real exponential function to the surreal numbers [19] and the resulting exponential field turns out to be elementarily equivalent to [13]. Berarducci and Mantova recently defined a derivation with respect to on the surreals [9], again with good model-theoretic properties [2]. In collaboration with Mantova, the authors constructed a surreal solution to the functional equation

which is a bijection of onto itself [6]. We call a hyperexponential and its functional inverse a hyperlogarithm.

The first goal of this paper is to extend the results from [6] to the construction of hyperexponentials of any ordinal force , together with their functional inverses . If is a successor ordinal, then satisfies the functional equation

Our second goal is to show that these hyperexponentials are “well-behaved” in the sense that they endow with the structure of a hyperserial field in the sense of [5].

1.1Motivation and background

Whereas it is natural to study surreal exponentiation and differentiation, it may seem more exotic to define and investigate the properties of surreal hyperexponentials and hyperlogarithms. In fact, the main motivation behind our work is a conjecture by the second author [26, p. 16] and a research program that was laid out in [1] for proving this conjecture. The ultimate goal is to expose the deep connections between two types of mathematical infinities: numerical infinities and growth rates at infinity. Let us briefly recall the rationale behind this connection.

Cantor's ordinal numbers provide us with a way to count beyond all natural numbers and to keep counting beyond the size of any set. However, ordinal arithmetic is rather poor in the sense that we have no subtraction or division and that addition and multiplication do not satisfy the usual laws of arithmetic, such as commutativity. We may regard Conway's surreal numbers as providing a calculus with Cantor's ordinal numbers which does extend the usual calculus with real numbers. In this sense, Conway managed to construct the ultimate framework for computations with numerical infinities.

Another source for computations with infinitely large quantities stems from the study of growth rates of real functions at infinity. The first major results towards a systematic asymptotic calculus of this kind are due to Hardy in [21, 22], based on earlier ideas by du Bois-Reymond [15, 16, 17]. Hardy defined an -function to be a function constructed from and the real numbers using the field operations, exponentiation, and logarithms. He proved that the germs of -functions at infinity form a totally ordered field. The framework of -functions is suitable for asymptotic analysis since we have an ordering for comparing the growth at infinity of any two such functions. This is often rephrased by saying that -functions have a regular growth at infinity.

Hardy also observed [21, p. 22] that “The only scales of infinity that are of any practical importance in analysis are those which may be constructed by means of the logarithmic and exponential functions.” In other words, Hardy suggested that the framework of -functions not only allows for the development of a systematic asymptotic calculus, but that this framework is also sufficient for all “practical” purposes. Alas, there are several “holes”. First of all, the framework is not closed under various useful operations such as functional inversion and integration. Secondly, the framework does not contain any functions of extremely fast or slow growth at infinity, like and , although such functions naturally appear in the analysis of certain algorithms. For instance, the best known algorithm for multiplying two polynomials of degree in runs in time ; see [23].

This raises the question how to construct a truly universal framework for computations with regular functions at infinity. Our next candidate is the class of transseries. A transseries is a formal object that is constructed from (with ) and the real numbers, using exponentiation, logarithms, and infinite sums. One example of a transseries is

Depending on conditions satisfied by their supports, there are different types of transseries. The first constructions of fields of transseries are due to Dahn and Göring [12] and Écalle [18]. More general constructions were proposed subsequently by the second author and his former student Schmeling [24, 25, 29]. Clearly, any -function is a transseries, but the class of transseries is also closed under integration and functional inversion, contrary to the class of -functions.

However, the class of transseries still does not contain any hyperexponential or hyperlogarithmic elements like or . In our quest for a truly universal framework for asymptotic analysis, we are thus lead to look beyond: a hyperseries is a formal object that is constructed from and the real numbers using exponentiation, logarithms, infinite sums, as well as hyperexponentials and hyperlogarithms of any force . The hyperexponentials and the hyperlogarithms are required to satisfy functional equations


where . For in Cantor normal form with , we also define


and we require that


It is non-trivial to construct fields of hyperseries in which these and several other technical properties (see section 4 below) are satisfied. This was first accomplished by Schmeling for hyperexponentials and hyperlogarithms of finite force . The general case was tackled in [14, 5].

The construction of general hyperseries relies on the definition of an abstract notion of hyperserial fields. Whereas the hyperseries that we are really after should actually be hyperseries in an infinitely large variable , abstract hyperserial fields potentially contain hyperseries that can not be written as infinite expressions in . In the present paper, we define hyperexponentials and hyperlogarithms on for all ordinals and show that this provides with the structure of an abstract hyperserial field. Moreover, any hyperseries in can naturally be evaluated at to produce a surreal number . The conjecture from [26, p. 16] states that, for a sufficiently general notion of “hyperseries in ”, all surreal numbers can actually be obtained in this way. We plan to prove this and the conjecture in a follow-up paper.

1.2General overview and summary of our new contributions

Our main goal is to define hyperexponentials for any ordinal and to show that is a hyperserial field for these hyperexponentials. Since our construction builds on quite some previous work, the paper starts with three sections of reminders.

In section 2, we recall basic facts about well-based series and surreal numbers. In particular, we recall that any surreal number can be regarded as a well-based series

with real coefficients . The corresponding group of monomials consists of those positive numbers that are of the form for certain subsets and of with .

Section 3 is devoted to the theory of surreal substructures from [4]. One distinctive feature of the class of surreal numbers is that it comes with a partial, well-founded order , which is called the simplicity relation. The Conway bracket can then be characterized by the fact that, for any sets and of surreal numbers with , there exists a unique -minimal number with . For many interesting subclasses of , it turns out that the restrictions of and to give rise to a structure that is isomorphic to . Such classes are called surreal substructures of and they come with their own Conway bracket .

In section 4, we recall the definition of hyperserial fields from [5] and the main results on how to construct such fields. One major fact from [5] on which we heavily rely is that the construction of hyperserial fields can be reduced to the construction of hyperserial skeletons. In the context of the present paper, this means that it suffices to define the hyperlogarithms only for very special, so called -atomic elements.

In the case when , the -atomic elements are simply the monomials in and the definition of the general logarithm on indeed reduces to the definition of the logarithm on : given , we write , where , and is infinitesimal, and we take . This very special case will be considered in more detail in section 5.

In the case when , the -atomic elements of are those elements such that is a monomial for every . The construction of on then reduces to the construction of on the class of -atomic numbers. This particular case was first dealt with in [6] and this paper can be used as an introduction to the more general results in the present paper.

For general ordinals , we say that is -atomic if is a monomial for every . The advantage of restricting ourselves to such numbers when defining hyperlogarithms is that only needs to verify few requirements with respect to the ordering. This makes it possible to define using the fairly simple recursive formula


where range over -atomic numbers with and ; see also (7.1).

In section 6, we prove that this definition is warranted and that the resulting functions satisfy the axioms of hyperserial skeletons from [5, Section 3]. Our proof proceeds by induction on and also relies on the fact that the class of -atomic numbers actually forms a surreal substructure of . Our main result is the following theorem:

Theorem 1.1. The definition (1.5) gives the structure of a confluent hyperserial skeleton in the sense of [5]. Consequently, we may uniquely extend to in a way that gives the structure of a confluent hyperserial field. Moreover, for each ordinal , the extended function is bijective.

Our final section 7 is devoted to further identities that illustrate the interplay between the hyperexponential and hyperlogarithmic functions and the simplicity relation on . We also prove the following more symmetric variant of (1.5):


where again range over the -atomic numbers with and . An interesting open question is whether there exists an easy argument that would allow us to use (1.6) instead of (1.5) as a definition of .

2Basic notions

2.1Ordered fields of well-based series

2.1.1Well-based series

Let be a (possibly class-sized) linearly ordered abelian group. We write for the class of functions whose support

is a well-based set, i.e. a set which is well-ordered with respect to the reverse order .

We see elements of as formal well-based series , where denotes the coefficient of in , for each . If , then we define to be the dominant monomial of . For , we let and we write . We say that a series is a truncation of and we write if . The relation is a well-founded partial order on with minimum .

By [20], the class is an ordered field under the pointwise sum

the Cauchy product

(where each sum has finite support), and where the positive cone is given by

The identification of with the formal series induces an ordered group embedding .

We next define the following asymptotic relations on :

The relation extends the ordering on . For non-zero we actually have (resp. , resp. ) if and only if (resp. , resp. ). We finally define

Series in , and are respectively called purely large, infinitesimal, and positive infinite.

2.1.2Well-based families

Let be a family in , We say that is well-based if

  1. is well-based, and

  2. is finite for all .

In that case, we may define the sum of by

If is another field of well-based series and is -linear, then we say that is strongly linear if for every well-based family in , the family is well-based, with

2.2Surreal numbers

2.2.1Surreal numbers and simplicity

We denote by the class of ordinal numbers. Following [19], we define to be the class of sign sequences

of ordinal length . The terms are called the signs of and we write for the length of . Given two numbers , we define

We call the simplicity relation on and note that is well-founded. See [4, Section 2] for more details about the interaction between and the ordered field structure of .

Recall that the Conway bracket is characterized by the fact that, for any sets and of surreal numbers with , there exists a unique -minimal number with . Conversely, given a number , we define

Then can canonically be written as

2.2.2Ordinals as surreal numbers

The structure contains an isomorphic copy of by identifying each ordinal with the constant sequence of length . We will write to state that is either an ordinal or the class of ordinals.

For , we write for the ordinal exponentiation of to the power and we define

If is a successor ordinal, then we define to be the unique ordinal with . We also define if is a limit ordinal. Similarly, if , then we set . Recall that every ordinal has a unique Cantor normal form

where , and with .

2.2.3Surreal numbers as well-based series

We define to be the class of positive numbers of the form for certain subsets and of with . Numbers in are called monomials. It turns out [11, Theorem 21] that the monomials form a subgroup of and that there is a natural isomorphism between and the ordered field . We will identify those two fields and thus see as a field of well-based series. The ordinal , seen as a surreal number, is the simplest element, or -minimum, of the class .

3Surreal substructures

3.1Surreal substructures

In [4], we introduced the notion of surreal substructures. A surreal substructure is a subclass of such that and are isomorphic. The isomorphism is unique and denoted by . Many important subclasses of that are relevant to the study of hyperserial properties of are surreal substructures. In particular, it is known that the following classes are surreal substructures:

If are surreal substructures, then the class is a surreal substructure with .


Given a subclass of and , we will write

so that and . We also write and .

If is a subclass of and are subsets of with , then the class

is called a cut in . If contains a unique simplest element, then we denote this element by and say that is a cut representation (of ) in . These notations naturally extend to the case when and are subclasses of with .

A surreal substructure may be characterized as a subclass of such that for all cut representations in , the cut has a unique simplest element [4, Proposition 4.7].

Let be a surreal substructure. Note that we have for all . Let and let be a cut representation of in . Then is cofinal with respect to in the sense that has no strict upper bound in and has no strict lower bound in [4, Proposition 4.11(b)].

Given numbers with , the number is the unique -maximal number with . We have . Let be a surreal substructure. Considering the isomorphism , we see that for all with , there is a unique -maximal element of with , and we have . In what follows, we will use this basic fact several times without further mention.

3.3Cut equations

Let be a subclass, let be a surreal substructure and be a function. Let be functions defined for cut representations in and such that are subsets of whenever is a cut representation in . We say that is a cut equation for if for all , we have

Elements in (resp. ) are called left (resp. right) options of this cut equation at .

We say that the cut equation is uniform if we have

whenever is a cut representation in . For instance, given , consider the translation on . By [19, Theorem 3.2], we have the following uniform cut equation for on :


We will need the following result from [4]:

Proposition 3.1. [4, Proposition 4.36] Let be surreal substructures. Let be a function from to the class of subsets of such that for with , the set is cofinal with respect to . For , let denote the class of elements of such that and are mutually cofinal. Let be a cut equation on that is extensive in the sense that

Let be strictly increasing with cut equation

Then induces an embedding for each element of .

3.4Convex partitions

One natural way to obtain surreal substructures is via convex partitions. If is a surreal substructure, then a convex partition of is a partition of whose members are convex subclasses of for the order . We may then consider the class of simplest elements (i.e. -minima) in each member of . Those elements are said -simple. For , we let denote the unique member of containing . By [4, Proposition 4.16], the class contains a unique -simple element, which we denote by . The function is a surjective non-decreasing function with .

Given , note that we have if and only if . For , we write . We have the following criterion to characterize elements of .

Proposition 3.2. [4, Lemma 6.5] An element of is -simple if and only if there is a cut representation of in with . Equivalently is -simple if and only if .

We say that is thin if each member of has a cofinal and coinitial subset. We then have:

Proposition 3.3. [4, Theorem 6.7 and Proposition 6.8] If is thin, then the class is a surreal substructure and has the following uniform cut equation:

3.5Function groups

A special type of thin convex partitions is that of partitions induced by function groups acting on surreal substructures. A function group on a surreal substructure is a set-sized group of strictly increasing bijections under functional composition. We see elements of as actions on and we sometimes write and instead of and , where .

For such a function group , the collection of classes

with is a thin convex partition of . We write . We have the uniform cut equation


Consider sets of strictly increasing bijections , then we say that is pointwise cofinal with respect to , and we write , if we have . We also define

It is easy to see that is a function group on and that we have if or . The relation trivially implies . If and , then we say that and are mutually pointwise cofinal and we write . We then have .

We write (resp. ) if we have (resp. ). We also write and instead of and .

Given a function group on , the relation defined by is a partial order on . We will frequently rely on the basic fact that is partially bi-ordered in the sense that

3.6Remarkable function groups

Each of the examples of surreal substructures from Subsection 3.1 can be regarded as the classes for actions of the following function groups acting on , or . For and , we define

Now consider

Then we have the following list of correspondences :

Generalizations of those function groups will allow us to define certain surreal substructures related to the hyperlogarithms and hyperexponentials on .

4Hyperserial fields

In this section, we briefly recall the definition of hyperserial fields from [5] and how to construct such fields from their hyperserial skeletons.

4.1Logarithmic hyperseries

Let be a formal, infinitely large indeterminate. The field of logarithmic hyperseries of [14] is the smallest field of well-based series that contains all ordinal real power products of the hyperlogarithms with . It is naturally equipped with a derivation and composition law .

Let be an ordinal. For each , we introduce the formal hyperlogarithm and define to be the group of formal power products with . This group comes with a monomial ordering that is defined by

We define to be the ordered field of well-based series . If are ordinals with , then we define to be the subgroup of of monomials with whenever . As in [14], we write

We have natural inclusions , hence natural inclusions .

Derivation on
The field is equipped with a derivation which satisfies the Leibniz rule and which is strongly linear. Write for all . The derivative of a logarithmic hypermonomial is defined by

So for all . For and , we will sometimes write .

Composition on
Assume that for a certain ordinal . Then the field is equipped with a composition that satisfies in particular:

The same properties hold for the composition if is replaced by . For , the map is injective, with image [14, Lemma 5.11]. For , we define to be the unique series in with .

4.2Hyperserial fields

Let be an ordered group. A real powering operation on is a law

of ordered -vector space on . Let be a field of well-based series with , let , and let be a function. For , we define to be the class of series with . We say that is a hyperserial field if


is a strongly linear morphism of ordered rings for each .


for all , , and .


for all , , and with .


for all ordinals , , and with .


The map extends to a real powering operation on .


for all .


for all ;

for all , and .

For each , we define the function . The skeleton of is defined to be the structure equipped with the real power operation from HF5.

We say that is confluent if for all with , we have

In particular is a confluent hyperserial field.

4.3Hyperserial skeletons

It turns out that each hyperlogarithm on a hyperserial field can uniquely be reconstructed from its restriction to the subset of -atomic hyperseries (here we say that is -atomic if for all ). One of the main ideas behind [14] is to turn this fact into a way to construct hyperserial fields. This leads to the definition of a hyperserial skeleton as a field with partially defined hyperlogarithms , which satisfy suitable counterparts of the above axioms HF1 until HF7.

More precisely, let be a field of well-based series and fix . A hyperserial skeleton on of force consists of a family of partial functions for , called (hyper)logarithms, which satisfy a list of axioms that we will describe now.

First of all, the domains on which the partial functions are defined should satisfy the following axioms:

Domains of definition:


, if is a non-zero limit ordinal;

, if is a successor ordinal.

It will be convenient to also define the class by

Consider an ordinal written in Cantor normal form where and . We denote by the partial function


It follows from the definition that for all , the class consists of those series for which and for all . We call such series -atomic.

Secondly, the hyperlogarithms with should satisfy the following axioms:

Axioms for the logarithm

Functional equation:








Surjective logarithm:


Axioms for the hyperlogarithms (for each with and )

Functional equation:

if is a successor ordinal.







Finally, for with , we also need the following axiom

Infinite products:


Note that and together imply , whence automatically holds. This will in particular be the case for (see Section 5).

In summary, we have:

Definition 4.1. [5, Definition 3.3] Given , we say that is a hyperserial skeleton of force if it satisfies , , , , and for all , as well as for all ordinals .

Assume that is a hyperserial skeleton of force . The partial logarithm extends naturally into a strictly increasing morphism , which we call the logarithm and denote by or [5, Section 4.1]. If satisfies , then this extended logarithm is actually an isomorphism [29, Proposition 2.3.8]. In that case, for any and , we define .


Definition 4.2. [5, Definition 3.5] Given a hyperserial skeleton of force and , we inductively define the notion of -confluence in conjunction with the definition of functions , as follows.

Let be such that is -confluent for all and let .

We say that is -confluent if each class contains a -atomic element; we then define to be this element.

This inductive definition is sound. Indeed, if and is -confluent for all , then the functions with are well-defined and non-decreasing. Thus, for , the collection of with forms a partition of into convex subclasses.

We say that is confluent if it is -confluent. If has force , then we say that is -confluent, or confluent, if is -confluent for all .

4.5Correspondence between fields and skeletons

Proposition 4.3. [5, Theorem 1.1] If is a confluent hyperserial skeleton, then there is a unique function with

such that is a confluent hyperserial field.

Assume now that is only a hyperserial skeleton of force and that is an ordinal with such that is -confluent. Let . By [5, Definition 4.11 and Lemma 4.12], the partial function naturally extends into a function that we still denote by . This extended function is strictly increasing, by‘ [5, Corollary 4.17]. If is a successor ordinal, then it satisfies the functional equation


by [5, Proposition 4.13]. For , we have a strictly increasing function obtained as a composition of functions with , as in (4.1). By [5, Proposition 4.7], we have


In a traditional transseries field , the transmonomials are characterized by the fact that, for any , we have


In particular, the logarithm is surjective as soon as is defined for all with . In hyperserial fields, similar properties hold for -atomic elements with respect to the hyperexponential , as we will recall now.

Given , let be a confluent hyperserial skeleton of force . By [5, Theorem 4.1], we have a composition . Given , the extended function is strictly increasing and hence injective. Consequently, has a partially defined functional inverse that we denote by .

The characterization (4.3) generalizes as follows:

Definition 4.4. [5, Definition 7.10] We say that is -truncated if

Given , we say that a series is -truncated if

For any , we write for the class of -truncated series in .

Proposition 4.5. [5, Corollary 7.21] For and , we have

In general, we have . Whenever is a successor ordinal, we even have


Let be a series such that is defined. By [5, Lemma 7.14], the series is -truncated if and only if

For , the axiom is therefore equivalent to the inclusion . For , there is a unique -maximal truncation of which is -truncated. By [5, Propositions 6.16 and 6.17], the classes


with form a partition of into convex subclasses. Moreover, the series is both the unique -truncated element and the -minimum of . If is defined, then we have the following simplified definition [5, Proposition 7.19] of the class :


The following shows that the existence of on is essentially equivalent to its existence on .

Proposition 4.6. [5, Corollary 7.24] Let and assume that for , the function is defined on . Then each hyperlogarithm for is bijective.

If Proposition 4.6 holds, then we say that is a (confluent) hyperserial field of force . Since every function is then a strictly increasing bijection , we obtain


for each ordinal with . By [5, Corollary 7.23], for all , we have


5The transseries field

Recall that is identified with the ordered field of well-based series . In this section, we describe, in the first level of our hierarchy, the properties of equipped with the Kruskal-Gonshor logarithm.

5.1Surreal exponentiation

In [19, Chapter 10], Gonshor defines the exponential function , relying on partial Taylor sums of the real exponential function. For and , write

We then have the recursive definition

We will sometimes write instead of . The function is a bijective morphism [19, Corollary 10.1, Corollary 10.3], which satisfies:

We define to be the functional inverse of , and we set . Given an ordinal , we understand that still stands for the -th ordinal power of from section 2.2.2 and warn the reader that does not necessarily coincide with .

Together, the above facts imply that satisfies the axioms , , , and . Therefore, is a hyperserial skeleton of force . The extension of to from section 4.5 coincides with . It was shown in [13] that is an elementary extension of . See [28, 7, 8] for more details on and .

5.2 as a transseries field

Berarducci and Mantova identified the class of -atomic numbers as [9, Corollary 5.17] and showed that is -confluent [9, Corollary 5.11]. Thus is a confluent hyperserial skeleton of force . Thanks to [5, Theorem 1.1], it is therefore equipped with a composition law . See [29, 10] for further details on extensions of this composition law to exponential extensions of .

Berarducci and Mantova also proved [9, Theorem 8.10] that is a field of transseries in the sense of [24, 29], i.e. that satisfies the axiom of [29, Definition 2.2.1]. We plan to prove in subsequent work that satisfies a generalized version of .

6Hyperserial structure on

We have seen in section 5 that is a confluent hyperserial skeleton of force for . The aim of this section is to extend this result to any ordinal . More precisely, we will define a sequence of partial functions on such that for each ordinal , the structure is a confluent hyperserial skeleton of force , and coincides with Gonshor's logarithm.

6.1Remarkable group actions on

Assume for the moment that we can define and as bijective strictly increasing functions on for all ordinals . This is the case already for . Let us introduce several useful groups that act on , as well as several remarkable subclasses of .

Given an ordinal , we write and we consider the function groups

where , and act on . We also define

We write and for each . In the case when , note that

By Proposition 3.3 and the fact the set-sized function groups , , , and induce thin partitions of , we may define the following surreal substructures

Here we note that corresponds to the class of infinite monomials in and maps positive infinite numbers to their dominant monomial. Similarly, coincides with and maps to . In sections 6 and 7, we will prove the following identities.

The first and third identities imply in particular that the classes and from section 4 are in fact surreal substructures, when regarding as a hyperserial field.

6.2Inductive setting

For the definition of the partial hyperlogarithm , we will proceed by induction on . Let be an ordinal. Until the end of this section we make the following induction hypotheses:

Induction hypotheses

For , the partial hyperlogarithm is defined; we have and is a confluent hyperserial skeleton of force .

For with and for with , we have

For , the class is that of -atomic surreal numbers, i.e. .

These induction hypotheses require a few additional explanations. Assuming that holds, the partial functions with extend into strictly increasing bijections , by the results from section 4. Using (1.3), this allows us to define a strictly increasing bijection for any and we denote by its functional inverse. In particular, this ensures that the hypotheses and make sense.

Remark 6.1. In addition to the above induction hypotheses, we will implicitly assume that our hyperlogarithms for are always defined by (6.1) below. In particular, our construction of is not relative to any potential construction of the preceding hyperlogarithms with that would satify the induction hypotheses , , and . Instead, we define one specific family of functions that satisfy our requirements, as well as the additional identities listed in Section 6.1.

Proposition 6.2. The axioms , and hold for .

Proof. Section 5 shows that holds. Consider with . On , we have , hence . It follows that we have on for all . This implies that holds. Finally, is valid because of the relation .

Proposition 6.3. Let be a limit ordinal and assume that , , and hold for all . Then , , and hold.

Proof. The statement follows immediately by induction. Towards , note that we have by (and thus ) and for all . By [4, Proposition 6.28], we have . So holds.

By for all , we need only justify that is -confluent to deduce that holds. For , by , there are a and a with . We deduce that , thus . This concludes the proof.

From now on, we assume that , , and are satisfied for and we define

The remainder of the section is dedicated to the definition of and the proof of the inductive hypotheses I, I, and I for . In combination with Propositions 6.2 and 6.3, this will complete our induction and the proof of Theorem 1.1.

6.3Defining the hyperlogarithm

Recall that we have by . In particular is a surreal substructure. Consider . The skeleton is a confluent hyperserial skeleton of force by . So for , (4.7) and yield .

In view of and , the simplest way to define is via the cut equation:


Note the asymmetry between left and right options and (instead of ) for generic and . In Corollary 7.4 below, we will derive a more symmetric but equivalent cut equation for , as promised in the introduction. For now, we prove that (6.1) is warranted and that , , and hold.

Proposition 6.4. The function is well-defined on and, for , we have


Proof. We prove this by induction on . Let such that holds for all . Let and . We have or , so or yields

For , we have and , whence

for all . Hence,

We clearly have . Finally,

so . This shows that is defined and

Since , it follows that

By induction, this proves for all .

Proposition 6.5. The axiom holds.

Proof. Let with . Since is a surreal substructure, there is a with and . If , then we have by . If , then we have by . We cannot have both and , so this proves that . Therefore holds.

Proposition 6.6. The axiom holds.

Proof. The rightmost options in (6.1) directly yield .

Proposition 6.7. The axiom holds.

Proof. Let and write . Let with . We have and so . Moreover is positive infinite. The number is strictly simpler than , so does not lie in the cut which defines in (6.1). Therefore, there is an or an and an ordinal with or . Consider the first case. We have for a certain . So and

For with , we have so . We deduce that for all such . It follows that for all . In the second case, we directly get . This proves that we always have . In other words , whence holds.

Proposition 6.8. If is a successor ordinal, then the cut equation (6.1) is uniform.

Proof. Let be a cut representation in and write . For , we have so . For , we have by . Since , it follows that . We may thus define the number

In order to show that (6.1) is uniform, we need to prove that , for any choice of the cut representation . We will do so by proving that and .

Recall that is cofinal with respect to and that is strictly increasing. Consequently, we have

Given , there is an with . By , we have for all , so . This proves that lies in the cut defining as per (6.1), whence .

Conversely, in order to prove that , it suffices to show that lies in the cut

Let and let be -maximal with . We have , whence , by . If , then , so yields and . Otherwise , so yields . This proves that .

Let and let be -maximal with . As above, if , then so yields , whence . Otherwise so yields . Hence and we conclude by induction.

6.4Functional equation

In this subsection we derive , under the assumption that is a successor ordinal. We start with the following inequality.

Lemma 6.9. If , then we have on .

Proof. For , there are and with . We have

on by (4.2). Note that , so yields

whence .

Let . Since is a surreal substructure, we may consider the -atomic number

We claim that . Assume that and write . We have

Assume now that . The function is strictly increasing with . Therefore

so . Since , the cut equation (6.1) for yields


Given , we have and . We deduce that

Moreover, by definition, we have

so . Symmetric arguments yield . Lemma 6.9 implies that , whence . We get , whence . Thus lies in the cut defining in (6.2), so . This proves our claim that


We now derive .

Proposition 6.10. For , we have .

Proof. We prove this by induction on . Let be such that the result holds on . By (6.3), we have

Let and range in and respectively. Proposition 6.8 and our induction hypothesis yield:

On the other hand, we have

In order to conclude that , it remains to show that and that . The first inequality holds because is a set of infinitesimal numbers. An easy induction shows that for all . The second inequality follows, because is strictly increasing on . This completes our inductive proof.

Combining our results so far, we have proved that is a hyperserial skeleton of force .


We next prove that is -confluent.

Lemma 6.11. If is a non-zero limit ordinal, then the function groups and are mutually pointwise cofinal. In particular, we have and .

Proof. For and , we have since . We have

whereas yields

Therefore . For , there is with . By (4.2), we have

which proves the inequality .

Lemma 6.12. For each , any -minimal element of is -atomic.

Proof. Let denote the class of numbers that are -minimal in . Any such -minimal number is also -minimal in , hence -atomic. Thus is defined on . It is enough to prove that is closed under in order to obtain that .

Consider , and recall that we have


Assume for contradiction that is not -minimal in . So there is a with . This implies that lies outside the cut defining , so is larger than a right option of (6.4) or smaller than a left option of (6.4).

Assume first that . So there is an with . We have so there is an with

The function is nondecreasing, so is nondecreasing as well. So . But


This contradicts the -minimality of .

Now consider the other case when . In particular, must be larger than a right option of (6.4). Symmetric arguments imply that we cannot have for some . So there must exist a with . If is a limit ordinal, then so Lemma 6.11 yields , whence . If is a successor ordinal, then there is a with , so

and Proposition 6.10 yields . In both cases, we thus have . For any integer , we deduce that

This contradicts the fact that lies in .

We have shown that the cases and both lead to a contradiction. Consequently, is -minimal in and we conclude that , as claimed.

Corollary 6.13. is -confluent.

Proof. We already know that is -confluent by . Recall that is well-founded, so each class for contains a -minimal element. Lemma 6.12 therefore implies that is -confluent.

The corollary implies that is a confluent hyperserial skeleton of force . Moreover, the class is that of -minima and thus -minima in the convex classes

for . In other words, we have . In order to conclude that is a surreal substructure, we still need to prove that the convex partition is thin. This will be done at the end of section 6.6 below.

Proposition 6.14. The cut equation (6.1) is uniform.

Proof. Let be a cut representation in and write . We have

By (4.6), this shows that

In particular, the number

is well-defined, with . As in the proof of Proposition 6.8, we have , whence . We conclude that the cut equation (6.1) is uniform.


We have shown that is a hyperserial skeleton of force . In order to prove that has force , it remains to prove that every -truncated number has a hyperexponential . This is the purpose of this subsection.

Proposition 6.15. We have , and has the following cut equation on :


Proof. We prove the result by induction on . Let such that is defined on with the given equation. We will first show that the number


is well-defined. We will then prove that .

Let and . If , then by the definition of . So . Otherwise, we have , whence by definition of , so . So we always have

We also have , so . This proves that . It remains to show that

Note that , so by the definition of , we have


Hence , which completes the proof that is well-defined.

Let us now prove that . Note that by Proposition 3.2. First assume that is a limit ordinal. Lemma 6.11 yields , so we may write

By (4.6), for the classes that and are mutually cofinal and coinitial. Moreover, we have for all , by our hypothesis on . Hence, Proposition 6.14 and (4.6) imply

Note that , so . Now . We also have


(by Lemma 6.11)
(by (6.7))

So . Since , the inequality follows from Proposition 3.2. Finally, we have by definition that , so . This proves that , so .

Assume now that is a successor ordinal. For all , the sets , , and are mutually cofinal. So we can rewrite (6.6) as

As in the limit case, Proposition 6.14 yields

Let . There is an with . Since , we have

In particular . We saw in (6.7) that , whence . We also obtain the inequalities

in a similar way as in the limit case.

We conclude that holds in general. It follows by induction that the formula for is valid. In particular is surjective.

With Proposition 6.15, we have completed the proof of . By (4.7), we have for all . Given , we also deduce from (4.6) that the set is cofinal and coinitial in . The convex partition defined by is thus thin. By Proposition 3.3, the class is a surreal substructure with uniform cut equation


For , we have , so . We deduce that the following equivalent is equivalent to (6.8):


6.7End of the inductive proof

We now prove , and Theorem 1.1.

Lemma 6.16. If is a limit ordinal, then we have on .

Proof. Let . We have , so (4.8) yields

We deduce that so by Lemma 6.11.

Proposition 6.17. For with and , we have on , i.e. holds.

Proof. Throughout this proof, we consider inequalities and equalities of functions on . Write and where and . We have

If , then , so yields , whence . Assume that . If is a successor ordinal, then there is with . By , we have . So . We conclude by noting that . If is a limit ordinal, then so by Lemma 6.16. It follows that for , we have . An easy induction on yields the result.

Proposition 6.18. is the class of -atomic numbers, i.e. holds.

Proof. Let . By Corollary 6.13, the simplest element of is -atomic. Since , we deduce that .

Conversely, given , we have . Now , so by , there are and with . Hence, , and . We conclude that .

In particular, the class is a surreal substructure. We have proved , and , so we obtain the following by induction:

Theorem 6.19. The field is a confluent hyperserial skeleton of force .

Combining this with Propositions 4.3 and 4.6, we obtain Theorem 1.1. Let us finally show that contains only one -atomic element.

Proposition 6.20. The number is the only -atomic element in . For all , there is with .

Proof. The number lies in for all , so it is -atomic. For , the number is an ordinal. As a sign sequence, the number is followed by a string containing only minuses [2, Lemma 2.6]. Since the sequences and are strictly increasing and strictly decreasing respectively, the classes and are respectively cofinal and coinitial in . Thus for , there is with , whence .

7Remarkable identities

In this section, we give various identities regarding the function groups introduced in Section 6.1. In what follows, is a non-zero ordinal and .

7.1Simplified cut equations for and

Given , let if is a successor ordinal and if is a limit ordinal. In this subsection, we will derive the following simplified cut equations for on and on :


For all , the set contains only -atomic numbers, so (7.3) is indeed a cut equation of the form .

Remark 7.1. The changes with respect to (6.1) and (6.5) lie in the occurrence of instead of in (7.2) and the (related) absence of the left option in (7.4). So (7.2) and (7.4) give lighter sets of conditions than those in (6.1) and (6.5) to define and . This seemingly meager simplification will be crucial in further work. Indeed, combined with Proposition 3.1, this allows one to determine large classes of numbers with .

First note that the cut equations (7.1) and (7.3) if they hold are uniform (see [6, Remark 1]). Moreover, we claim that (7.1,7.2) are equivalent and that (7.3,7.4) are equivalent. Indeed, recall that for a thin convex partition of a surreal substructure and any cut representation in , one has

For and the classes and are mutually cofinal by (4.6). Similarly, and are mutually coinitial. By Lemma 6.11, the classes and are mutually cofinal. So it is enough to prove that (7.1) and (7.3) are valid cut equations for and respectively.

Lemma 7.2. If is a successor ordinal, then the identities (7.1) and (7.3) hold.

Proof. Let and set

We have so in view of (6.1), it is enough to prove that to conclude that . Let . If , then the inequality entails whence and . Otherwise, we have , so , and . It is enough to prove that . Recall that

by (3.1). We see that for all . We have so . Thus . So (7.1) holds.

Now let and set

By uniformity of (7.1), we have

whence . Conversely, and , so . We have . Since , this yields . This proves that lies in the cut defining . We conclude that , hence (7.3) holds.

We now assume that is a limit ordinal. For , define

Lemma 7.3. For all , we have


Proof. We prove the result by induction on . Let be such that (7.5), (7.6), (7.7) and (7.8) hold for all with .

For and , we have . We have by definition of if and by definition of if . This proves that is defined.

Let and . If , then we have by definition of . Since and , we have for all We deduce that . If , then by definition of . Since , we obtain . This proves that is defined.

Since (7.7) and (7.8) hold on , we have

By (6.9), this yields , so (7.7) holds for .

From (7.7), we get . By Proposition 6.14 and our assumption that (7.8) holds on , we have

Recall that . Therefore it suffices to show that lies in the cut to conclude that and thus that . Now so and . We have , where . Since is a limit ordinal, Lemma 6.11 implies that , so . This completes the proof that .

Corollary 7.4. The identities (7.1), (7.2), (7.3), and (7.4) all hold.

Proof. It is enough to prove (7.1) and (7.3). The identity (7.3) follows from (7.7) and (7.8). In order to obtain (7.1), we consider , set , and we show that . Since (7.3) is uniform, we have

We have because , and because on . Since , we deduce that .

Remark 7.5. The simplified cut equations for can be viewed as alternative definitions for those functions, since they hold inductively on their domain of definition. It is unclear how to develop our theory directly upon these alternative definitions. In particular, does there exists a direct way to see that the cut equation (7.2) is warranted, and that the corresponding function satisfies and ?

7.2Identities involving and .

Proposition 7.6. Defining as in Section 6.1, we have .

Proof. Let . We have by [5, Proposition 7.22]. Recall that for all . Now by definition of , so and . By definition of , we conclude that .

Assume that is a successor ordinal. Then we have by (4.4), so the functions and are both strictly increasing bijections from onto .

Lemma 7.7. Assume that is a successor ordinal. Then for , we have on .

Proof. Let us abbreviate . We prove the lemma by induction on . Let with

whenever is strictly simpler than . We let denote generic elements of and we note that . By (6.8), we have

Recall that is a successor ordinal. Since (4.2) holds for all , the sets and are mutually cofinal and coinitial. Moreover for all and , so

By (3.1), we have

The numbers and are -truncated so lies in the cut

We deduce that . The result follows by induction.

Lemma 7.8. If is a successor ordinal, then we have on . Consequently, .

Proof. The set is pointwise cofinal in . So is pointwise cofinal in . For , there is such that . We have

We deduce that on , whence .

7.3Identities involving and .

Lemma 7.9. If is a successor ordinal, then for we have

Proof. This can be seen as a converse to the proof of the identity (6.3). We proceed by induction on . Let be such that the relation holds on . By (6.3), we have

We conclude by induction.

Noting that on , the previous relation further generalizes as follows.

Proposition 7.10. Assume that is a successor ordinal and let . Then


Proof. We proceed by induction. Let be such that

for all strictly simpler with respect to the product order . For , let be the function on and let . By (3.1) and (3.2), we have

By (7.1), Lemma 7.7 and (3.1), we have:

We deduce that

It is enough to prove that to conclude that . Towards this, fix an with . Lemma 7.9 yields

We conclude by induction that (7.9) holds.

Remark 7.11. For , we have , and . Therefore we can see as a system of fractional and real iterates of the hyperexponential function on . The previous proposition shows that the action of those iterates on -atomic numbers reduces to translations, modulo the parametrization . In particular, one can compute the functional square root of on in terms of sign sequences using the material from [3].

Proposition 7.12. If is a successor ordinal, then .

Proof. For , we have . By Lemma 7.9, it follows that . This implies that , so is -simple.

Conversely, consider such that is -simple. We have and , whence . We obtain , which proves that .

Proposition 7.13. We have .

Proof. Let . So . By Proposition 3.1, the number is simplest in

Since , we have so . We deduce that , so is -simple. Conversely, let . By Proposition 3.1 the number is simplest in . Since , we have so . We deduce that is -simple.

Corollary 7.14. If is a successor ordinal, then .



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simplest number between and 4

field of well-based series with real coefficients over 5

support of a series 5


truncation of 5





and 5

series with 5

series with 5

series with and 5

class of ordinals 6

simplicity relation 6

ordinal exponentiation with base at 6

if is a successor ordinal and if is a limit ordinal 6

for 6

the surreal substructure 7

class of -simple elements 8

projection 8

class of numbers with 8

comparison between sets of strictly increasing bijections 9

function group generated by 9

and are mutually pointwise cofinal 9

translation 9

homothety 9

power function 9

function group 9

function group 9

function group 9

function group 9

function group 9

field of logarithmic hyperseries 10

group of logarithmic hypermonomials of force 10

field of logarithmic hyperseries of force 10

unique series in with 10

hyperlogarithm function 11

class of -atomic series 11

functional equation 12

asymptotics axiom 12

monotonicity axiom 12

regularity axiom 12

infinite products axiom 12

class of series with 13

-atomic element of 13

hyperexponential function 14

class of -truncated series 14

series with 14

-maximal -truncated truncation of 14

axiom for transseries fields [29, Definition 2.2.1] 15

function group 16

function group 16

function group 16

function group 16

structure of -simple elements 16

structure of -simple elements 16

structure of -simple elements 16

structure of -simple elements 16