Surreal numbers as hyperseries

At the end of the 19-th century, two theories emerged for computations with infinitely large quantities. The first one was due to du Bois-Reymond [19, 20, 21], who developed a “calculus of infinities” to deal with the growth rates of functions in one real variable at infinity. The second theory of “ordinal numbers” was proposed by Cantor [13] as a way to count beyond the natural numbers and to describe the sizes of sets in his recently introduced set theory.

Du Bois-Reymond's original theory was partly informal and not to the taste of Cantor, who misunderstood it [25]. The theory was firmly grounded and further developed by Hausdorff and Hardy. Hausdorff formalized du Bois-Reymond's “orders of infinity” in Cantor's set-theoretic universe [24]. Hardy focused on the computational aspects and introduced the differential field of logarithmico-exponential functions [28, 29]: such a function is constructed from the real numbers and an indeterminate

(that we think of as tending to infinity) using the field operations, exponentiation, and the logarithm. Subsequently, this led to the notion of a Hardy field [12].

As to Cantor's theory of ordinal numbers, Conway proposed a dramatic generalization in the 1970s. Originally motivated by game theory, he introduced the proper class

of surreal numbers [14], which simultaneously contains the set

of all real numbers and the class

of all ordinals. This class comes with a natural ordering and arithmetic operations that turn

into a non-Archimedean real closed field. In particular,

are all surreal numbers, where

stands for the first infinite ordinal.

Conway's original definition of surreal numbers is somewhat informal and draws inspiration from both Dedekind cuts and von Neumann's construction of the ordinals:

The notation

is called Conway's bracket. Conway proposed to consider

as the simplest number between

and

. Indeed, it turns out that one may define a partial ordering

with

for any number

with

. This so-called simplicity relation has the additional property that any

can be written canonically as

One may regard

as the set of surreal numbers that were defined before

when using Conway's recursive definition. Conway's bracket is uniquely determined by the simplicity relation

and vice versa.

The ring operations on

are defined in a recursive way that is both very concise and intuitive: given

and

, we define

It is quite amazing that these definitions coincide with the traditional definitions when

and

are real, but that they also work for the ordinal numbers and beyond. Subsequently, Gonshor also showed how to extend the real exponential function to

[26] and this extension preserves all first order properties of

[16]. Simpler accounts and definitions of

can be found in [37, 9].

The theory of Hardy fields focuses on the study of growth properties of germs of actual real differentiable functions at infinity. An analogue formal theory arose after the introduction of transseries by Dahn and Göring [15] and, independently, by Écalle [22, 23]. Transseries are a natural generalization of the above definition of Hardy's logarithmico-exponential functions, by also allowing for infinite sums (modulo suitable precautions to ensure that such sums make sense). One example of a transseries is

In particular, any transseries can be written as a generalized series

with real coefficients

and whose (trans)monomials

are exponentials of other (generally “simpler”) transseries. The support

of such a series should be well based in the sense that it should be well ordered for the opposite ordering of the natural ordering

on the group of transmonomials

. The precise definition of a transseries depends on further technical requirements on the allowed supports. But for all reasonable choices, “the” resulting field

of transseries possesses a lot of closure properties: it is ordered and closed under derivation, composition, integration, and functional inversion [22, 30, 18]; it also satisfies an intermediate value property for differential polynomials [32, 3].

It turns out that surreal numbers and transseries are similar in many respects: both

and

are real closed fields that are closed under exponentiation and taking logarithms of positive elements. Surreal numbers too can be represented uniquely as Hahn series

with real coefficients

and monomials in a suitable multiplicative subgroup

. Any transseries

actually naturally induces a surreal number

by substituting

for

and the map

is injective [11].

But there are also differences. Most importantly, elements of

can be regarded as functions that can be derived and composed. Conversely, the surreal numbers

come equipped with the Conway bracket. In fact, it would be nice if any surreal number could naturally be regarded as the value

of a unique transseries

. Indeed, this would allow us to transport the functional structure of

to the surreal numbers. Conversely, we might equip the transseries with a Conway bracket and other exotic operations on the surreal numbers. The second author conjectured the existence of such a correspondence between

and a suitably generalized field of the transseries [32, page 16]; see also [2] for a more recent account.

Now we already observed that at least some surreal numbers

can be written uniquely as

for some transseries

. Which numbers and what kind of functions do we miss? Since a perfect correspondence would induce a Conway bracket on

, it is instructive to consider subsets

with

and examine which natural growth orders might fit between

and

One obvious problem with ordinary transseries is that there exists no transseries that grows faster than any iterated exponential

. Consequently, there exists no transseries

with

. A natural candidate for a function that grows faster than any iterated exponential is the first hyperexponential

, which satisfies the functional equation

It was shown by Kneser [33] that this equation actually has a real analytic solution on

. A natural hyperexponential

was constructed more recently in [8]. In particular,

More generally, one can formally introduce the transfinite sequence

of hyperexponentials of arbitrary strengths

, together with the sequence

of their functional inverses, called hyperlogarithms. Each

with

satisfies the equation

and there again exist real analytic solutions to this equation [38]. The function

does not satisfy any natural functional equation, but we have the following infinite product formula for the derivative of every hyperlogarithm

The traditional field

of transseries is not closed under hyperexponentials and hyperlogarithms, but it is possible to define generalized fields of hyperseries that do enjoy this additional closure property. Hyperserial grow rates were studied from a formal point of view in [22, 23]. The first systematic construction of hyperserial fields of strength

is due to Schmeling [38]. In this paper, we will rely on the more recent constructions from [17, 7] that are fully general. In particular, the surreal numbers

form a hyperserial field in the sense of [7], when equipped with the hyperexponentials and hyperlogarithms from [6].

Here again, there exists no transseries

with

. This cut has actually a natural origin, since any “tame” solution of the functional equation

lies in this cut. What is missing here is a suitable notion of “nested transseries” that encompasses expressions like

This type of cuts were first considered in [30, Section 2.7.1]. Subsequently, the second author and his former PhD student Schmeling developed an abstract notion of generalized fields of transseries [31, 38] that may contain nested transseries. However, it turns out that expressions like (1.2) are ambiguous: one may construct fields of transseries that contain arbitrarily large sets of pairwise distinct solutions to (1.1).

In order to investigate this ambiguity more closely, let us turn to the surreal numbers. The above cut

induces a cut

. Nested transseries solutions

to the functional equation (1.1) should then give rise to surreal numbers

with

and such that

are all monomials in

. In [5, Section 8], we showed that those numbers

actually form a class

that is naturally parameterized by a surreal number (

forms a so-called surreal substructure). Here we note that analogue results hold when replacing Gonshor's exponentiation by Conway's

-map

(which generalizes Cantor's

-map when

). This was already noted by Conway himself [14, pages 34–36] and further worked out by Lemire [34, 35, 36]. Section 6 of the present paper will be devoted to generalizing the result from [5, Section 8] to nested hyperseries.

Besides the two above types of superexponential and nested cuts, no other examples of “cuts that cannot be filled” come naturally to our mind. This led the second author to conjecture [32, page 16] that there exists a field

of suitably generalized hyperseries in

such that each surreal number can uniquely be represented as the value

of a hyperseries

. In order to prove this conjecture, quite some machinery has been developed since: a systematic theory of surreal substructures [5], sufficiently general notions of hyperserial fields [17, 7], and definitions of

on the surreals that give

the structure of a hyperserial field [8, 6].

Now one characteristic property of generalized hyperseries in

should be that they can uniquely be described using suitable expressions that involve

, real numbers, infinite summation, hyperlogarithms, hyperexponentials, and a way to disambiguate nested expansions. The main goal of this paper is to show that any surreal number can indeed be described uniquely by a hyperserial expression of this kind in

. This essentially solves the conjecture from [32, page 16] by thinking of hyperseries in

as surreal numbers in which we replaced

. Of course, it remains desirable to give a formal construction of

that does not involve surreal numbers and to specify the precise kind of properties that our “suitably generalized” hyperseries should possess. We intend to address this issue in a forthcoming paper.

Other work in progress concerns the definition of a derivation and a composition on

. Now Berarducci and Mantova showed how to define a derivation on

that is compatible with infinite summation and exponentiation [10]. In [4, 1], it was shown that there actually exist many such derivations and that they all satisfy the same first order theory as the ordered differential field

. However, as pointed out in [2], Berarducci and Mantova's derivation does not obey the chain rule with respect to

. The hyperserial derivation that we propose to construct should not have this deficiency and therefore be a better candidate for the derivation on

with respect to

1.2.Outline of our results and contributions

In this paper, we will strongly rely on previous work from [5, 17, 7, 8, 6]. The main results from these previous papers will be recalled in Sections 2, 3, and 4. For the sake of this introduction, we start with a few brief reminders.

The field of logarithmic hyperseries

was defined and studied in [17]. It is a field of Hahn series

in the sense of [27] that is equipped with a logarithm

, a derivation

, and a composition

. Moreover, for each ordinal

, it contains an element

such that

for all

and all ordinals

. Moreover, if the Cantor normal form of

is given by

with

, then we have

The derivation and composition on

satisfy the usual rules of calculus and in particular a formal version of Taylor series expansions.

In [7], Kaplan and the authors defined the concept of a hyperserial field to be a field

of Hahn series with a logarithm

and a composition law

, such that various natural compatibility requirements are satisfied. For every ordinal

, we then define the hyperlogarithm

of strength

. We showed in [6] how to define bijective hyperlogarithms

for which

has the structure of a hyperserial field. For every ordinal

, the functional inverse

is called the hyperexponential of strength

The main aim of this paper is to show that any surreal number

is not just an abstract hyperseries in the sense of [6], but that we can regard it as a hyperseries in

. We will do this by constructing a suitable unambiguous description of

in terms of

, the real numbers, infinite summation, the hyperexponentials, and the hyperlogarithms.

for some ordinary transseries

, then the idea would be to expand

as a linear combination of monomials, then to rewrite every monomial as an exponential of a transseries, and finally to recursively expand these new transseries. This process stops whenever we hit an iterated logarithm of

In fact, this transserial expansion process works for any surreal number

. However, besides the iterated logarithms (and exponentials) of

, there exist other monomials

such that

is a monomial for all

. Such monomials are said to be log-atomic. More generally, given

, we say that

is -atomic if

for all

. We write

for the set of such numbers. If we wish to further expand an

-atomic monomial

as a hyperseries, then it is natural to pick

such that

is not

-atomic, to recursively expand

, and then to write

Unfortunately, the above idea is slightly too simple to be useful. In order to expand monomials as hyperseries, we need something more technical. In Section 5, we show that every non-trivial monomial

has a unique expansion of exactly one of the two following forms:

where

with

, and where

lies in

. Moreover, if

then it is imposed that

, and that

cannot be written as

where

, and

After expanding

in the above way, we may pursue with the recursive expansions of

and

as hyperseries. Our next objective is to investigate the shape of the recursive expansions that arise by doing so. Indeed, already in the case of ordinary transseries, such recursive expansions may give rise to nested expansions like

Expansions of the forms (1.5) and (1.6) are said to be well-nested and ill-nested, respectively. The axiom T4 for fields of transseries in [38] prohibits the existence of ill-nested expansions. It was shown in [10] that

satisfies this axiom T4.

The definition of hyperserial fields in [6] does not contain a counterpart for the axiom T4. The main goal of section 4 is to generalize this property to hyperserial fields and prove the following theorem:

Now there exist surreal numbers for which the above recursive expansion process leads to a nested expansion of the form (1.5). In [5, Section 8], we proved that the class

of such numbers actually forms a surreal substructure. This means that

is isomorphic to

for the restriction

. In particular, although the nested expansion (1.5) is inherently ambiguous, elements in

are naturally parameterized by surreal numbers in

The main goal of Section 6 is to prove a hyperserial analogue of the result from [5, Section 8]. Now the expansion (1.5) can be described in terms of the sequence

. More generally, in Section 6 we the define the notion of a nested sequence in order to describe arbitrary nested hyperserial expansions. Our main result is the following:

In Section 7, we reach the main goal of this paper, which is to uniquely describe any surreal number as a generalized hyperseries in

. This goal can be split up into two tasks. First of all, we need to specify the hyperserial expansion process that we informally described above and show that it indeed leads to a hyperserial expansion in

, for any surreal number. This will be done in Section 7.2, where we will use labeled trees in order to represent hyperserial expansions. Secondly, these trees may contain infinite branches (also called paths) that correspond to nested numbers in the sense of Section 6. By Theorem 1.2, any such nested number can uniquely be identified using a surreal parameter. By associating a surreal number to each infinite branch, this allows us to construct a unique hyperserial description in

for any surreal number and prove our main result:

2.Ordered fields of well-based series

2.1.Well-based series

Let

be a totally ordered (and possibly class-sized) abelian group. We say that

is well-based if it contains no infinite ascending chain (equivalently, this means that

is well-ordered for the opposite ordering). We denote by

the class of functions

whose support

is a well-based. The elements of

are called monomials and the elements in

are called terms. We also define

We see elements

as formal well-based series

where

for all

. If

, then

is called the dominant monomial of

. For

, we define

and

. For

, we sometimes write

. We say that a series

is a truncation of

and we write

. The relation

is a well-founded partial order on

with minimum

where each sum

is finite. The class

is actually an ordered field, whose positive cone

is defined by

We also write

whenever

and

. If

are non-zero, then

(resp.

, resp.

) if and only if

(resp.

, resp.

Series in

and

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

2.2.Well-based families

is another field of well-based series and

-linear, then we say that

is strongly linear if for every well-based family

, the family

is well-based, with

2.3.Logarithmic hyperseries

The field

of logarithmic hyperseries plays an important role in the theory of hyperseries. Let us briefly recall its definition and its most prominent properties from [17].

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

By what precedes,

is an ordered field of well-based series. If

are ordinals with

, then we define

to be the subgroup of

of monomials

with

whenever

. As in [17], we define

The field

is equipped with a derivation

which satisfies the Leibniz rule and which is strongly linear: for each

, we first define the logarithmic derivative of

. The derivative of a logarithmic hypermonomial

is next defined by

Finally, this definition extends to

by strong linearity. Note that

for all

. For

and

, we will sometimes write

Assume that

for a certain ordinal

. Then the field

is also equipped with a composition

that satisfies:

The same properties hold for the composition

, when

is replaced by

. For

, the map

is injective, with range

[17, Lemma 5.11]. For

, we define

to be the unique series in

with

3.Surreal numbers as a hyperserial field

3.1.Surreal numbers

of “signs”

indexed by arbitrary ordinals

. We will write

for the domain of such a sequence and

for its value at

. Given sign sequences

and

, we define

Conway showed how to define an ordering, an addition, and a multiplication on

that give

the structure of a real closed field [14]. See [5, Section 2] for more details about the interaction between

and the ordered field structure of

. By [14, Theorem 21], there is a natural isomorphism between

and the ordered field of well-based series

, where

is a certain subgroup of

. We will identify those two fields and thus regard

as a field of well-based series with monomials in

The partial order

contains an isomorphic copy of

obtained by identifying each ordinal

with the constant sequence

of length

. We will write

to specify that

is either an ordinal or the class of ordinals. The ordinal

, seen as a surreal number, is the simplest element, or

-minimum, of the class

For

, we write

and

for the non-commutative ordinal sum and product of

and

, as defined by Cantor. The surreal sum and product

and

coincide with the commutative Hessenberg sum and product of ordinals. In general, we therefore have

and

For

, we write

for the ordinal exponentiation of base

. Gonshor also defined an exponential function on

with range

. One should not confuse

with

, which yields a different number, in general. We define

where

and

with

. The ordinals

are called the exponents of

and the integers

its coefficients. We write

(resp.

) if each exponent

of the Cantor normal form of

satisfies

(resp.

are ordinals, then we write

, we write

if there exists an

with

, and we write

if both

and

hold. The relation

is a quasi-order on

. For

with

and

, we have

. In particular, we have

for all

is a successor, then we define

to be the unique ordinal with

. We also define

is a limit. Similarly, if

, then we write

3.2.Hyperserial structure on

We already noted that Gonshor constructed an exponential and a logarithm on

and

, respectively. We defined hyperexponential and hyperlogarithmic functions of all strengths on

in [6]. In fact, we showed [6, Theorem 1.1] how to construct a composition law

with the following properties:

Note that the composition law on

also satisfies

(but not

), with each occurrence of

being replaced by

3.3.Hyperlogarithms

For

, we write

for the function

, called the hyperlogarithm of strength

. By

and

, this is a strictly increasing bijection. We sometimes write

for

. We write

for the functional inverse of

, called the hyperexponential of strength

Let

and write

for the coefficient in

in the Hahn series representation of

. There is a unique infinitesimal number

with

. We write

for the natural logarithm on

. The function defined by

is called the logarithm on

. This is a strictly increasing morphism

which extends

. It also coincides with the logarithm on

that was defined by Gonshor.

3.4.Atomicity

Given

, we write

for the class of numbers

with

for all

. Those numbers are said to be

-atomic and they play an important role in this paper. Note that

and

for all

, in view of (3.1). There is a unique

-atomic number [6, Proposition 6.20], which is the simplest positive infinite number

Each hyperlogarithmic function

with

is essentially determined by its restriction to

, through a generalization of (3.3). More precisely, for

, there exist

and

with

. Moreover, the family

is well-based, and the hyperlogarithm

is given by

3.5.Hyperexponentiation

Given

with

, we write

for the class of

-truncated numbers. Note that

. We will sometimes write

when

. For

, there is a unique

-maximal truncation

which is

-truncated. By [7, Proposition 7.17], the classes

with

form a partition of

into convex subclasses. Moreover, the series

is both the unique

-truncated element and the

-minimum of the convex class containing

. We have

The formulas (3.3) and (3.4) admit hyperexponential analogues. For all

, there is a

with

. For any such

, there is a family

with

such that

is well-based and

4.Surreal substructures

In [5], we introduced the notion of surreal substructure. A surreal substructure is a subclass

such that

and

are isomorphic. The isomorphism

is unique and denoted by

. For the study of

as a hyperserial field, many important subclasses of

turn out to be surreal substructures. In particular, given

, it is known that the following classes are surreal substructures:

We will prove in Section 6 that certain classes of nested numbers also form surreal substructures.

4.1.Cuts

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

, the cut

has a unique simplest element [5, Proposition 4.7].

Let

be a surreal substructure. Note that we have

for all

. Let

and let

be a cut representation of

. Then

is cofinal with respect to

in the sense that

has no strict upper bound in

and

has no strict lower bound in

[5, Proposition 4.11(b)].

4.2.Cut equations

Let

be a subclass, let

be a surreal substructure and

be a function. Let

be functions defined for cut representations in

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

for all cut representations

. For instance, given

, consider the translation

. By [26, Theorem 3.2], we have the following uniform cut equation for

Let

with

and set

. We have the following uniform cut equations for

and

[6, Section 8.1]:

4.3.Function groups

A function group

on a surreal substructure

is a set-sized group of strictly increasing bijections

under functional composition. We see elements

as actions on

and sometimes write

and

for

rather than

and

. We also write

for the functional inverse of

with

forms a partition of

into convex subclasses. For subclasses

, we write

. An element

is said to be

-simple if it is the simplest element inside

. We write

for the class of

-simple elements. Given

, we also define

to be the unique

-simple element of

. The function

is a non-decreasing projection of

onto

. The main purpose of function groups is to define surreal substructures:

Note that for

, we have

if and only if

. We have the following criterion to identify the

-simple elements inside

, then we say that

is pointwise cofinal with respect to

. For

, we also write

instead of

and

Given a function group

, we define a partial order

. We will frequently rely on the elementary fact that this ordering is compatible with the group structure in the sense that

Given a set

of strictly increasing bijections

, we define

to be the smallest function group on

that is generated by

, i.e.

4.4.Remarkable function groups

The examples of surreal substructures from the beginning of this section can all be obtained as classes

-simplest elements for suitable function groups

that act on

, or

, as we will describe now. Given

and

, we first define

Now the action of

yields the surreal substructure

as class of

-simplest elements. All examples from the beginning of this section can be obtained in a similar way:

Let

and

. We will need a few inequalities from [6]. The first one is immediate by definition and the fact that

. The others are [6, Lemma 6.9, Lemma 6.11, and Proposition 6.17], in that order:

5.Well-nestedness

In this section, we prove Theorem 1.1, i.e. that each number is well-nested. In Section 5.1 we start with the definition and study of hyperserial expansions. We pursue with the study of paths and well-nestedness in Section 5.2.

The general idea behind our proof of Theorem 1.1 is as follows. Assume for contradiction that there exists a number

that is not well-nested and choose a simplest (i.e.

-minimal) such number. By definition,

contains a so-called “bad path”. For the ill-nested number

from (1.6), that would be the sequence

thereby contradicting the minimality assumption on

. In order to make this idea work, we first need a series of “deconstruction lemmas” that allow us to affirm that

is indeed simpler than

; these lemmas will be listed in Section 5.3. We will also need a generalization

of the relation

that was used by Berarducci and Mantova to prove the well-nestedness of

as a field of transseries; this will be the subject of Section 5.4. We prove Theorem 1.1 in Section 5.5. Unfortunately, the relation

does not have all the nice properties of

. For this reason, Sections 5.4 and 5.5 are quite technical.

5.1.Hyperserial expansions

Recall that any number can be written as a well-based series. In order to represent numbers as hyperseries, it therefore suffices to devise a means to represent the infinitely large monomials

. We do this by taking a hyperlogarithm

of the monomial and then recursively applying the same procedure for the monomials in this new series. This procedure stops when we encounter a monomial in

Technically speaking, instead of directly applying a hyperlogarithm

to the monomial, it turns out to be necessary to first decompose

as a product

and write

as a hyperexponential (or more generally as the hyperlogarithm of a hyperexponential). This naturally leads to the introduction of hyperserial expansions of monomials

, as we will detail now.

Formally speaking, hyperserial expansions can be represented by tuples

. By convention, we also consider

to be a hyperserial expansion of the monomial

; this expansion is represented by the tuple

is tail-atomic since

is log-atomic. Now

is a hyperserial expansion of type II and we have

. Hence

is a hyperserial expansion.

Let

, so

. We may further expand each monomial in

. We clearly have

. We claim that

. Indeed, if we could write

for some

and

, then

and

would both be monomials, which cannot be. Note that

, so

is a hyperserial expansion of type I. We also have

where

is tail expanded. Thus

is a hyperserial expansion. Note finally that

is a hyperserial expansion. We thus have the following “recursive” expansion of

Proof. We first prove the result for

, by induction with respect to the simplicity relation

. The

-minimal element of

, which satisfies (5.2) for

and

. Consider

such that the result holds on

. By [6, Proposition 6.20], the monomial

is not

-atomic. So there is a maximal

with

, and we have

by our hypothesis.

If there is no ordinal

such that

, then we have

. So setting

and

, we are done. Otherwise, let

be such that

. We cannot have

by definition of

. So there is a unique ordinal

and a unique natural number

such that

and

. Note that

. We must have

: otherwise,

where

and

are monomials. We deduce that

and

. Note that

, and

, so

. We deduce that

. The induction hypothesis yields a hyperserial expansion

. Since

is log-atomic, we must have

and

. If

, then

, since

. Thus

is a hyperexponential expansion of type II. If

, then likewise

and thus

is a hyperexponential expansion of type I. This completes the inductive proof.

Now let

and set

. If

is tail-atomic, then there are

and

with

. Applying the previous arguments to

, we obtain elements

with

and

, or an ordinal

with

. Then

is a hyperserial expansion. If

is not tail-atomic, then we have

is a hyperserial expansion of type I.

Proof. We first prove

). Assume that

-atomic. Assume for contradiction that

and let

denote the least non-zero term in the Cantor normal form of

. Since

, we have

is a monomial. But

where

is a monomial: a contradiction. So

. If

then we are done. Otherwise

, so we must have

, whence

. Conversely, assume that

, and that

. If

, then then for all

, we have

where

, so

is a monomial, whence

. If

, then for all

, we have

, whence

. This proves

Now assume that

. So

-atomic by

). If

then we conclude that

. If

, then let

denote the least non-zero term in its Cantor normal form. We have

and

, so

Proof. Since

, we must have

. By Lemma 5.4, we have a hyperserial expansion

. Since

is log-atomic, we have

, whence

and

. So

. We have

so by Lemma 5.5(a), we have

. It follows that

is a hyperserial expansion. But then

and Lemma 5.5(a) imply that

. The condition that

now gives

, whence

is a successor and

for a certain

, as claimed.

where

, and

. Assume for contradiction that we can write

as a hyperserial expansion of type I with

. Note in particular that

, so

is log-atomic. We have

, then

, and

is not tail-atomic. But

, where

, so

is tail-atomic: a contradiction. Hence

. Note that

and

are both the least term of

. It follows that

, and

Now

, so

and thus

. In particular

. Taking hyperexponentials on both sides of (5.4), we may assume without loss of generality that

or that the least exponents

and

in the Cantor normal forms of

resp.

differ. If

, then we decompose

where

and

. Since

, applying Lemma 5.5(a) twice (for

and

) gives

and

, whence

. But then

, where

by Lemma 5.5(a). So

: a contradiction. Assume now that

. Lemma 5.5(a) yields both

and

, which contradicts (5.4).

Taking

and

, this proves that no two hyperserial expansions of distinct types I and II can be equal. Taking

with

, this proves that no two hyperserial expansions

of type I with

can be equal.

The two remaining cases are hyperserial expansions of type II and hyperserial expansions

and

of type I with

. Consider a monomial

with the hyperserial expansions

of type II. As above we have

, and

. We deduce that

, so the expansions coincide.

We may assume without loss of generality that

. Assume for contradiction that

. Taking hyperexponentials on both sides of (5.6), we may assume without loss of generality that

or that the least exponents

and

in the Cantor normal forms of

resp.

differ. On the one hand, Lemma 5.5(a) yields

. Note in particular that

, since

. On the other hand, if

, then Lemma 5.5(a) yields

; if

, then

. Thus (5.6) is absurd: a contradiction. We conclude that

. Finally

yields

, so the expansions are identical.

Proof. Assume for contradiction that

. In particular

. Since

, there is

with

, so

: a contradiction.

5.2.Paths and subpaths

We call

the length of

and we write

. We say that

is infinite if

and finite otherwise. We set

. For

, we define

By Lemma 5.8, those cases are mutually exclusive so

is well-defined. For

, we say that

is a path in

Example 5.9. Let us find all the paths in the monomial

of Example 5.3. We have a representation (5.3) of

as a hyperseries

Note that the paths which cannot be extended into strictly longer paths are those whose last value is a real number or

Infinite paths occur in so-called nested numbers that will be studied in more detail in Section 6.

Remark 5.11. The above definition extends the former definitions of paths in [30, 38, 10]. More precisely, a path

with with

(whence

) for all

, corresponds to a path for these former definitions. The validity of the axiom T4 for

means that those paths are good. With Theorem 1.1, we will extend this result to all paths.

Proof. Let

and let

be a path in

. If there is an ordinal

with

, then the hyperserial expansion of

, so

and

otherwise. If there is an ordinal

with

, then the hyperserial expansion of

and

Assume now that

. If

is not tail-atomic, then hyperserial expansion of

. If

is tail-atomic, then the hyperserial expansion of

for a certain log-atomic

. In both cases,

is a path in some monomial in

, whence

and

, by the previous argument.

Let

be a finite path and let

be a path with

. Then we define

to be the path

of length

Proof. By Lemma 5.12, we have

. If

has a hyperserial expansion of the form

, then

must be a path in

. So

is non-zero and thus

. It follows that

is a path in

. Otherwise, let

be the hyperserial expansion of

. If

is a path in

, then it is a path in

as above. Otherwise, it is a path in

. Assume that

. If

, then we have

and

is a path in

. If

, then

where

is a hyperserial expansion, so

is a path in

. Assume now that

, so

, and

. We must have

so there are

and

with

and

. We have

where

is a hyperserial expansion, so

is a path in

Proof. We prove this by induction on

, for any number

. We consider

, and a fixed path

with

Assume that

. We have

and

. Assume that

for certain

, and

. Let

be the hyperserial expansion of

. If

, then

and the hyperserial expansion of

. Therefore

is a subpath in

. If

, then the hyperserial expansion of

. Therefore

is a subpath in

. Finally, if

is not tail-atomic, then

is a subpath in

, where

is the sign of

Now assume that

, and that the result holds strictly below

. We have

where

is a subpath in

by the previous argument. We have

for a certain

, so

is a path in

. The induction hypothesis on

implies that

is a subpath in

Assume now that

and that the result holds strictly below

. Write

. By (3.7), there exist

, and

with

and

Assume for contradiction that there is a

with

. We must have

, so there are a number

and an ordinal

with

. We have

. By Lemma 5.12, this contradicts the fact that

. So by Lemma 5.4, there exist

and

with

, and

. Since

, we must have

so there are a number

and an ordinal

with

(note that

whenever

). Thus

is a monomial with hyperserial expansion

. There is no path in

of length

, so

must be a path in

. We deduce that

is a path in

. Consequently,

is a path in

with

. Applying

times Lemma 5.14, we deduce that

is a subpath in

, hence in

. Consider a path

with

for a certain

. Applying the induction hypothesis for

and

in the roles of

and

, the path

is a subpath in

. Therefore

is a subpath in

. We deduce as in the case

that

is a subpath in

Proof. Let

be a hyperserial expansion. The condition

implies

, whence

is also a hyperserial expansion. In particular

is a subpath in

Assume that

shares a subpath with

. In other words, there is a path

which has a common subpath with

. The path

must be infinite, so by Lemma 5.15, it shares a subpath with

. Let us prove by induction on

that

shares a subpath with

. Assuming that this holds for

, we note that

-atomic, hence

-atomic. So

shares a subpath with

by Lemma 5.14 and the induction hypothesis. We conclude by induction that

shares a subpath with

Suppose conversely that

shares a subpath with

. By induction on

, it follows from Lemma 5.15 that

shares a subpath with

. Applying Lemma 5.14 to

, we conclude that

shares a subpath with

5.3.Deconstruction lemmas

In this subsection, we list several results on the interaction between the simplicity relation

and various operations in

Proof. The condition

yields

. We have

by Lemma 5.23. The identity

implies that

, whence

by Lemma 5.21. Consequently,

, by Lemma 5.22. Since

, we may apply Lemma 5.22 to

and

to obtain

. We conclude using the transitivity of

5.4.Nested truncation

In [10, Section 8], the authors prove the well-nestedness axiom

for

by relying on a well-founded partial order

that is defined by induction. This relation has the additional property that

In this subsection, we define a similar relation

that will be instrumental in deriving results on the structure of

. However, this relation does not satisfy

for all

where

is a sequence of relations that are defined by induction on

, as follows. For

, we set

, if

or if there exist decompositions

with

and

. Assuming that

has been defined, we set

if we are in one of the two following configurations:

Warning 5.26. Taking

in the first configuration, we see that

extends

. However, the relation

is neither transitive nor anti-symmetric. Furthermore, as we already noted above, we do not have

Proof. Let

and

. Assume for contradiction that

and

. Assume first that

, so

. Then

. Let

be such that

. Since

, we deduce that

, whence

. Modulo replacing

, it follow that we may assume without loss of generality that

for some

and some monomial

On the one hand,

is not

-truncated, so there are

and

with

and

. We may choose

for certain

and

, so

. On the other hand,

-truncated, so we have

We deduce that

. If

is a successor, then choosing

, we obtain

, so

: a contradiction. Otherwise,

by [7, (2.4)], where

. Thus

, whence

: a contradiction.

We now treat the general case. By a similar argument as above, we may assume without loss of generality that

. Assume that

. Since

is not

-truncated, there exists a

with

, whence

. But

-truncated, so

. In particular

, so our hypothesis

implies that

: a contradiction.

Assume now that

. As in the first part of the proof, there are

and

with

and

. Recall that

for sufficiently large

. Take

and

such that

Then

. If

is a successor, then choosing

yields

, which contradicts the fact that

and

are infinitesimal. So

is a limit. Writing

, we have

. As in the first part of the proof, we obtain

, so

. In view of (5.11), we also obtain

, so

: a contradiction.

Proof. Assume that

. Let

and let

be its functional inverse in

. We have

by (4.10, 4.11), whence

. Furthermore,

, so

We want to prove that

. By (5.12), it is enough to prove that there is a

such that the inequality

holds on

Proof. We prove this by induction on

with

. Let

be an infinite path in

. Assume that

. If

, then we have

is a path in

. Otherwise, there are

and

with

and

. Then

for certain

and

with

. We must have

. If

is a path in

, then it is a path in

. Otherwise, it is a path in

, so

is a subpath in

, hence in

We now assume that

where

and that the result holds for all

and

with

and

. Assume first that

is in Configuration I, and write

Then we can write

like in the case when

. If

is a path in

, then it is a path in

. So we may assume that

is a path in

. Note that we have

. Setting

, we observe that

, whence

is the hyperserial expansion of

. If

is a path in

, then it is a path in

Suppose that

is not a path in

. Assume first that

, so

, and

is a path in

. Then Lemma 5.14 implies that

is a subpath in

, so

is a subpath in

. Otherwise, consider the hyperserial expansion

. Since

is not a path in

, it must be a path in

. The number

-atomic, so we must have

and

. There are

and

such that

. Therefore

. It follows by Corollary 5.17 that

shares a subpath with

, whence so does

Let

. Recall that

, so

. Now (4.8) implies that

, so

. The function

is non-decreasing, so

. But

, so the induction hypothesis yields that

, and thus

, shares a subpath with

. We deduce with Lemma 5.15 that

shares a subpath with

, hence with

Note that we also have

. We may again assume that

is a path in

. Write

, where

, and

. Then

where

. We deduce by induction that

shares a subpath with

. By Lemma 5.15, it follows that

shares a subpath with

, hence with

. This concludes the proof.

Proof. i. If

, then we have

, so

. Let

with

. Since

and

are monomials, we have

, whence

. Our assumption that

also implies

. Hence

. Now

shares a subpath with

, by Lemma 5.15. Since

, Proposition 5.29 next implies that

shares a subpath with

. Using Lemma 5.14, we conclude that

shares a subpath with

, and hence with

ii. Let

with

. It is enough to prove that

shares a subpath with

. Since

, and

are monomials, we have

. Let

, so that

. In particular, we have

. Moreover

, so using Lemma 5.15 and Proposition 5.29, we deduce in the same way as above that

shares a subpath with

. If

, then

is the hyperserial expansion of

, so

shares a subpath with

. If

, then the hyperserial expansion of

must be of the form

, since otherwise

would have at least two elements in its support. We deduce that

shares a subpath with

and that the hyperserial expansion of

. Therefore

shares a subpath with

iii. We assume that

is not

-truncated whereas

and

. If

, then we must have

, which means that

or that

and

. But then

: a contradiction.

Assume that

. By Lemma 5.27, we may assume without loss of generality that

. The assumption on

and the fact that

imply that

is non-zero. Write

and

. Note that

must be infinitesimal since

is not

-truncated. Thus

is also infinitesimal. By Lemma 5.27, we deduce that

. We have

, so

, since

and

are both

-truncated. Since

is not

-truncated, there is an ordinal

with

. If

, then

, because

-truncated. Thus

. If

, then

, because

and

are

-truncated. Now

, since

. We again deduce that

In both cases, we have

where

, so

shares a subpath with

, by Proposition 5.29. It follows by Corollary 5.17 that

shares a subpath with

5.5.Well-nestedness

We now prove that every number is well-nested. Throughout this subsection,

will be an infinite path inside a number

. At the beginning of Section 5.2 we have shown how to attach sequences

, etc. to this path. In order to alleviate notations, we will abbreviate

, and

for all

We start with a technical lemma that will be used to show that the existence of a bad path

implies the existence of a bad path in a strictly simpler number than

We first treat the case when

. Note that

since it contains a subpath, so

. From our assumption that

and the fact that

, we deduce that

. Hence

and (5.15). Note that (5.19) and (5.20) follow immediately from the other assumptions on

. If

then

. If

, then

, since

and

. Hence

by Lemma 5.21 and

by Lemmas 5.19 and 5.22. Finally,

by Lemma 5.20, so (5.21) holds in general. Recall that

is a subpath in

, but that it shares no subpath with

. In view of (5.20), we deduce (5.16) from Lemma 5.30(i) and (5.17) from Lemma 5.30(ii). Combining (5.16), (5.17) and (5.20) with the relation

, we finally obtain (5.18).

Let

and assume that (5.15–5.21) hold for all

. We shall prove (5.15–5.21) if

, as well as (5.19) and (5.21). Recall that

Proof of Theorem 1.1. Assume for contradiction that the theorem is false. Let

be a

-minimal ill-nested number and let

be a bad path in

. Let

be the smallest bad index in

. As in Lemma 5.31, we define

, and

for all

. We may assume that

, otherwise the number

is ill-nested and satisfies

: a contradiction.

Assume for contradiction that there is a

such that

is ill-nested. Set

is ill-nested and

otherwise. If

, then

cannot share a subpath with

, so

by Lemma 5.30, and

is ill-nested. In general, it follows that

is ill-nested. Let

be a bad path in

and set

. Then we may apply Lemma 5.31 to

, and

in the roles of

, and

. Since

, this yields an ill-nested number

: a contradiction.

Therefore the numbers

are well-nested. Since

is bad for

, one of the four cases listed in Definition 5.10 must occur. We set

By construction, we have

. Furthermore

shares a subpath with

, so there exists a bad path

. We have

by Lemma 5.27. If Definition 5.10(4) occurs, then we must have

is written as in (5.13) with

in the role of

and

. Otherwise,

is as in (5.13) for

. Setting

, it follows that we may apply Lemma 5.31 to

and

in the roles of

and

. We conclude that there exists an ill-nested number

: a contradiction.

6.Surreal substructures of nested numbers

In the previous section, we have examined the nature of infinite paths in surreal numbers and shown that they are ultimately “well-behaved”. In this section, we work in the opposite direction and show how to construct surreal numbers that contain infinite paths of a specified kind. We follow the same method as in [5, Section 8].

Let us briefly outline the main ideas. Our aim is to construct “nested numbers” that correspond to nested expressions like

Nested expressions of this kind will be presented through so-called coding sequences

. Once we have fixed such a coding sequence

, numbers

of the form (6.1) need to satisfy a sequence of natural inequalities: for any

with

, we require that

Numbers that satisfy these constraints are said to be admissible. Under suitable conditions, the class Ad of admissible numbers forms a convex surreal substructure. This will be detailed in Sections 6.1 and 6.2, where we will also introduce suitable coordinates

The notation (6.1) also suggests that each of the numbers

should be a monomial. An admissible number

is said to be nested if this is indeed the case. The main result of this section is Theorem 1.2, i.e. that the class

of nested numbers forms a surreal substructure. In other words, the notation (6.1) is ambiguous, but can be disambiguated using a single surreal parameter.

6.1.Coding sequences

Taking

for all

, we obtain a reformulation of the notion of coding sequences in [5, Section 8.1]. If

is a coding sequence and

, then we write

Proof. Let

. We have

because

is a good index for

. We have

and

by the definition of hyperserial expansions. If

and

, then we have

because

by the definition of paths. Lemma 5.27 also yields

. This proves the conditions

) and

) for coding sequences. Assume that

. Then by the definition of hyperserial expansions, we have

and

is not tail-atomic. Assume that

. Then

. We have

where

and

is not tail-atomic. This implies that

is not log-atomic, so

. Thus

) is valid.

Assume now for contradiction that there is an

with

for all

. By

), we have

for all

, and the sequence

is non-increasing, hence eventually constant. Let

with

for all

. For

, we have

. Therefore

-atomic: a contradiction. We deduce that

) holds as well.

We next fix some notations. For all

with

, we define partial functions

and

The domains of these functions are assumed to be largest for which these expressions make sense. We also write

We note that on their respective domains, the functions

, and

are strictly increasing if

, and

, respectively, and strictly decreasing in the contrary cases. We will write

and

for the partial inverses of

and

. We will also use the abbreviations

Proof. Let us prove the lemma by induction on

. The result clearly holds for

. Assuming that

is well defined, let

be minimal such that

. Note that we have

, so

where

. Applying

to the inequality

Both in the cases when

and when

, it follows that

is bounded from below by the hyperexponential

of a number. Thus

is well defined and so is each

for

. If

, then we have

and

Both in the cases when

and when

, it follows that

is bounded from below by the hyperexponential

of a number, so

is well defined and so is each

for

6.2.Admissible sequences

Note that we do not ask that

be a hyperserial expansion, nor even that

be a monomial. For the rest of the section, we fix a coding sequence

. We write

for the class of

-admissible numbers. If

, then the definition of

implicitly assumes that

is well defined for all

. Note that if

is admissible, then so is

for

. We denote by

the corresponding class of

-admissible numbers.

The main result of this subsection is the following generalization of [5, Proposition 8.2]:

As a consequence of this last proposition and [5, Proposition 4.29(a)], the class

is a surreal substructure if and only if

is admissible.

We use the notations from Section 6.1. We claim that

is admissible. Indeed for

, set

Proof. For

, we write

and

have the same sign. Let us prove by induction on

that

is defined and that

. Since this implies that

, that

, and that

, this will yield

The result follows from our hypothesis if

. Assume now that

and let us prove that

. Let

We have

, so

is defined. Moreover

. Since

, we deduce that

, whence in particular

. This concludes the proof.

Proof. Let

be minimal with

. We thus have

. We have

and

where

. We deduce by induction using Lemma 5.28 that

6.3.Nested sequences

In this subsection, we assume that

is admissible. For

we say that a

-admissible number

-nested if we have

for all

. We write

for the class of

-nested numbers. For

we simply say that

-nested and we write

Note that the inclusion

always holds. In [5, Section 8.4], we gave examples of nested and admissible non-nested sequences in the case of transseries, i.e. with

for all

. We next give an example in the hyperserial case.

Example 6.11. We claim that the sequence

from Example 6.6 is nested. Indeed, let

and

. We have

for a certain

with

. Let us check that the conditions of Definition 6.4 are satisfied for

Secondly, let

. We want to prove that

. We have

for a certain

. Then

by the previous paragraph. Now

Finally, we claim that

. This is immediate since the dominant term

is positive infinite, so

. Therefore

is nested.

A crucial feature of nested sequences is that they are sufficient to describe nested expansions. This is the content of Theorem 6.15 below.

Proof. Recall that

. By Corollary 6.8, we have

,. So we may assume without loss of generality that

We claim that

. Assume for contradiction that

and write

accordingly. Then Corollary 5.6 implies that

, in which case we define

, or

for some ordinal

and

for some

. Therefore

, so

. This implies that

Recall that

. Assume that

, so

. Since

-atomic, we also have

. Let

be minimal with

. We have

and

. In particular, the number

is log-atomic. If

, this contradicts the fact that

. If

, then

implies

But then

is not a monomial: a contradiction. Assume now that

. So

and

. But then

is not defined: a contradiction. We conclude that

, or if

and

is not tail-atomic, then our claim yields the result. Assume now that

and that

is tail-atomic where

, and

is a hyperserial expansion. Then the hyperserial expansion of

We next show that

. If

, then

, and we conclude with Lemma 6.7 that

. Assume for contradiction that

. Since

is log-atomic, we must have

. By the definition of coding sequences, this implies that

and

. So

, whence

, and

. In particular the number

-atomic, hence tail-atomic. Since

, the claim in the second paragraph of the proof, applied to

, gives

. But then also

: a contradiction.

We pursue with two auxiliary results that will be used order to construct a infinite path required in the proof of Theorem 6.15 below.

Proof. By Lemma 5.16, it is enough to find such a path in

. Write

. Assume first that

, so

and

. If

is not tail-atomic, then the hyperserial expansion of

and

is the dominant term of

for some

. Then the path

with

and

satisfies

. If

is tail-atomic, then there exist

and

such that the hyperserial expansion of

. Let

be a term in

with

. Then the path

with

and

satisfies

Assume now that

. In view of (3.6), we recall that there are an ordinal

and a number

with

is a limit ordinal, then by Lemma 6.12, we have a hyperserial expansion

. Let

and set

and

, so that

is a path in

. By Lemma 5.15, there is a subpath in

, hence also a path

, with

. So

. If

is a successor ordinal, then we may choose

for a certain

. By Lemma 6.12, we have a hyperserial expansion

. As in the previous case, there is a path

with

, whence

Proof. This is immediate if

. Assume that the result holds at

and pick a corresponding path

with

(resp.

). Note that the dominant term

(resp.

) lies in

by Lemma 6.7. Moreover

is a term of

(resp.

). By the previous lemma, there is a path

with

, so

satisfies the conditions.

Proof. Assume for contradiction that this is not the case. This means that the set

of indices

such that we do not have

is infinite. We write

where

. Fix

and let

. Let

such that

We claim that we can extend

to a path

with

and such that

is a bad index in

. Indeed, in view of Definition 6.4 for

, the relation (6.2) translates into the following three possibilities:

Consider a

and the path

. So

is a finite path with

. Thus there exists a path

which extends

with

, where

is a bad index in

. Repeating this process iteratively for

, we construct a path

that extends

and such that

is a bad index in

. At the limit, this yields an infinite path

that extends each of the paths

. This path

has a cofinal set of bad indices, which contradicts Theorem 1.1. We conclude that there is a

such that

is nested.

Proof. Note that

. The result thus follows from Corollary 6.8 and the assumption that

is nested.

Proof. The proof is similar to the proof of Lemma 5.31. We have

and we must have

since

. If follows from the deconstruction lemmas in Section 5.3 that

. This proves the result in the case when

, then we have

by assumption and we have shown above that

. We have

and

is a monomial, so (6.3) yields

. This deals with the case

. In addition, we have

because

and

. Let us show that

, then this follows from the facts that

and

. If

and

, then

. If

, then

and

, so

Assume now that

and that the induction hypothesis holds for all smaller

. We have

Since

is nested, we immediately obtain

, whence

as above. Since

and

is nested, we have

. Using (6.5), (6.4), and the decomposition lemmas, we observe that the relation

is equivalent to

For

and

, we have

by Lemma 6.16. We may thus consider the strictly increasing bijection

We will prove Theorem 1.2 by proving that the function group

generates the class

, i.e. that we have

. We first need the following inequality:

Proof. It is enough to prove the result for

. Assume that

. Let

and set

, so that

, then

and

is strictly increasing. So we only need to prove that

, which reduces to proving that

. Let

be the dominant term of

. Our assumption that

is nested gives

, whence

. We deduce that

. Lemma 5.27 implies that

-truncated.

is a strictly positive term. We deduce that

, whence

. The other cases when

or when

are proved similarly, using symmetric arguments.

Proof. By Proposition 4.1, the class

is a surreal substructure, so it is enough to prove the equality. We first prove that

Assume for contradiction that there are an

and a

, which we choose minimal, such that

cannot be written as

where

is a hyperserial expansion. Set

and

Assume that this is proved and set

. The first condition and Lemma 6.16 yield

and the relations

and

. The second and third condition, together with Lemma 6.17, imply

. The first condition also implies that

: a contradiction. Proving the existence of

and

is therefore sufficient.

and

, then

and

satisfy (6.7). Assume now that

and that

, whence

. If

then

and

satisfy (6.7). Assume therefore that

. This implies that there exist

and

with

By the definition of coding sequences, there is a least index

with

, so

We have

and

. So by Corollary 5.6, we must have

for a certain

and

for a certain

. Note that

. Recall that

and

, so

. The case

cannot occur for otherwise

Conversely, consider

and set

. So there are

and

with

. Let

. By Lemma 6.18, there exist

with

and

, whence

. Since

is strictly monotonous, we get

. The numbers

and

are monomials, so

. Therefore

In view of Theorem 6.19, Lemma 6.18, and Proposition 4.1, we have the following parametrization of

Proof. By [5, Lemma 4.5] and since the function

is strictly monotonous, it is enough to prove that

. By induction, we may also restrict to the case when

. So assume that

. Recall that

by Lemma 6.9. Since

, we deduce with Lemma 5.25 that

. It follows using the decomposition lemmas that

Proof. We have

by definition of

. So we only need to prove that

. Consider

. Since

is nested, the number

-admissible, so we need only justify that

. Since

-admissible, we have

. But

-nested, so

for a certain term

. We deduce that

, whence

Assume for contradiction that

and write

where

and

. Note that

: otherwise

and

would be zero for all

, thereby contradicting Definition 6.1(e). By Corollary 5.6, we must have

for a certain ordinal

and

for a certain

. Consequently,

. If

, then the condition

implies

, which leads to the contradiction that

. If

, then

, whence

: a contradiction.

Proof. Assume that

, or

and

. In particular, if

-admissible, then

, so

. For

, it follows that

-admissible if and only if

-admissible. Let

be the class of

-admissible numbers, for each

. We have

by the previous remarks, and

is admissible. For

, we have

, so

is nested. We deduce that

, that is, we have a strictly increasing bijection

. It is enough to prove that for

with

, we have

. Proceeding by induction on

, we may assume without loss of generality that

. By [6, identity (6.3)], the function

has the following equation on

So it is enough to prove that

. Note that

and

where

. So

, whence

. This concludes the proof.

6.4.Pre-nested and nested numbers

Let

be a number. We say that

is pre-nested if there exists an infinite path

without any bad index for

. In that case, Lemma 6.2 yields a coding sequence

which is admissible due to the fact that

with the notations from Section 6. By Theorem 6.15, we get a smallest

such that

is nested. If

, then we say that

is nested. In that case, Theorem 6.19 ensures that the class

-nested numbers forms a surreal substructure, so

can uniquely be written as

for some surreal parameter

One may wonder whether it could happen that

. In other words: do there exist pre-nested numbers that are not nested? For this, let us now describe an example of an admissible sequence

such that the class

-nested numbers contains a smallest element

. This number

is pre-nested, but cannot be nested by Theorem 6.19. Note that our example is “transserial” in the sense that it does not involve any hyperexponentials.

Example 6.25. Let

be a nested sequence with

. Let

be the simplest

-nested number. We define a coding sequence

where

is an infinite monomial, so

-nested. In particular, the sequence

is admissible.

Assume for contradiction that there is a

-nested number

with

. Since

, we have

. Recall that

and

are purely large, so

. In particular

which contradicts the assumption that

-nested. We deduce that

is the minimum of the class

-nested numbers. In view of Theorem 6.19, the sequence

cannot be nested.

The above examples shows that there exist admissible sequences that are not nested. Let us now construct an admissible sequence

such that the class

-nested numbers is actually empty.

Example 6.26. We use the same notations as in Example 6.25. Define

and set

for all

. We claim that the coding sequence

is admissible. In order to see this, let

. Then

Since

-admissible (i.e.

-admissible), we deduce that

-admissible, whence

is admissible. Assume for contradiction that

is non-empty, and let

. Then

-nested, so

, whence

: a contradiction.

7.Numbers as hyperseries

Traditional transseries in

can be regarded as infinite expressions that involve

, real constants, infinite summation, exponentiation and logarithms. It is convenient to regard such expressions as infinite labeled trees. In this section, we show that surreal numbers can be represented similarly as infinite expressions in

that also involve hyperexponentials and hyperlogarithms. One technical difficulty is that the most straightforward way to do this leads to ambiguities in the case of nested numbers. These ambiguities can be resolved by associating a surreal number to every infinite path in the tree. In view of the results from Section 6, this will enable us to regard any surreal number as a unique hyperseries in

Remark 7.1. In the case of ordinary transseries, our notion of tree expansions below is slightly different from the notion of tree representations that was used in [30, 38]. Nevertheless, both notions coincide modulo straightforward rewritings.

7.1.Introductory example

In order to formalize the general recursive expansion process, it is more convenient to work with the unsimplified version of this expression

Introducing

as a notation for the “power” operator, the above expression may naturally be rewritten as a tree:

In the next subsection, we will describe a general procedure to expand surreal monomials and numbers as trees.

7.2.Tree expansions

In what follows, a tree

is a set of nodes

together with a function that associates to each node

an arity

and a sequence

of children; we write

for the set of children of

. Moreover, we assume that

contains a special element

, called the root of

, such that for any

there exist a unique

(called the height of

and also denoted by

) and unique nodes

with

, and

for

. The height

of the tree

is the maximum of the heights of all nodes; we set

if there exist nodes of arbitrarily large heights.

Given a class

, an -labeled tree is a tree together with a map

, called the labeling. Our final objective is to express numbers using

-labeled trees, where

Instead of computing such expressions in a top-down manner (from the leaves until the root), we will compute them in a bottom-up fashion (from the root until the leaves). For this purpose, it is convenient to introduce a separate formal symbol

for every

, together with the extended signature

We use

as a placeholder for a tree expression for

whose determination is postponed to a later stage.

Consider a

-labeled tree

and a map

. We say that

is an evaluation of

if for each node

one of the following statements holds:

We call

the value of

via

. We say that

is a value of

if there exists an evaluation of

with

Proof. This is straightforward, by applying the rules E1–E7 recursively (from the leaves to the root in the case of (a) and the other way around for (b)).

Although evaluations with a given end-value are unique for a fixed tree

, different trees may produce the same value. Our next aim is to describe a standard way to expand numbers using trees. Let us first consider the case of a monomial

. If

, then the standard monomial expansion of

is the

-labeled tree

with

and

. Otherwise, we may write

with

. Depending on whether

, we respectively take

and call

the standard monomial expansion of

. Let us next consider a general number

and let

be the ordinal size of its support. Then we may write

for a sequence

and a

-decreasing sequence

. For each

, let

be the standard monomial expansion of

. Then we define the

-labeled tree

and call it the standard expansion of

. Note that the height of

is at most

, there exists a unique evaluation

, and

Now consider two trees

and

with respective labelings

and

. We say that

refines

and there exist evaluations

and

such that

for all

and

whenever

. Now assume that

for some evaluation

. Then we say that

is a tree expansion of

if for every

with

, the subtree

with root

refines the standard expansion of

. In particular, a tree expansion

of a number

with

always refines the standard expansion of

Proof. Given

, we say that an

-labeled tree

is -settled if

for all nodes

of height

. Let us show how to construct a sequence

-labeled tree expansions of

such that the following statements hold for each

We take

such that

and

. Setting

, the conditions S1, S2, S3, and S4 are naturally satisfied.

Assume now that

has been constructed and let us show how to construct

. Let

be the subset of

of nodes

of level

with

. Given

, let

be the standard expansion of

and let

be the unique evaluation of

. We define

to be the tree that is obtained from

when replacing each node

by the tree

Since each tree

is of height at most

, the height of

is finite. Since

is clearly

-settled, this proves S1. We define an evaluation

by setting

for any

and

for any

and

(note that

is well defined since

for all

). We have

, so S2 holds for

. By construction,

and the evaluations

and

coincide on

; this proves S3. Finally, let

be a tree expansion of

with labels in

and let

be the unique evaluation of

with

. Then

refines

, so

coincides with

. Let

. Since

is a tree expansion of

, the subtree

with root

refines

, whence

. Moreover,

, so

coincides with

. Altogether, this shows that

refines

Having completed the construction of our sequence, we next define a

-labeled tree

and a map

by taking

and by setting

and

for any

and

such that

. By construction, we have

and

refines

for every

We claim that

is a tree expansion of

. Indeed, consider a node

of height

with

. Then

and

, since

-settled. Consequently, the subtree of

with root

refines the standard expansion of

. Since

refines

, it follows that the subtree of

with root

also refines the standard expansion of

. This completes the proof of our claim.

It remains to show that

is the unique tree expansion of

with labels in

. So let

be any tree expansion of

with labeling

. For every

, it follows from S4 that

. Moreover, since

-settled,

coincides with both

and

on those nodes in

that are of height

. Consequently,

and

coincides with

. Since every node in

has finite height, we conclude that

7.3.Hyperserial descriptions

From now on, we only consider tree expansions with labels in

, as in Lemma 7.3. Given a class

of nested numbers as in Section 6, it can be verified that every element in

has the same tree expansion. We still need a notational way to distinguish numbers with the same expansion.

Let

be a pre-nested number. By Theorem 6.15, we get a smallest

such that

is nested. Hence

for the class

-nested numbers. Theorem 6.19 implies that there exists a unique number

with

. We call

the nested rank of

and write

. By Corollary 6.23, we note that

for all

. Given an arbitrary infinite path

in a number

, there exists a

such that

has no bad indices for

(modulo a further increase of

, we may even assume

to be nested). Let

. We call

the nested rank of

, where we note that the value of

does not depend on the choice of

Let

be the tree expansion of a number

and let

be the evaluation with

. An infinite path in

is a sequence

of nodes in

with

and

for all

. Such a path induces an infinite path

: let

be the indices with

; then we take

for each

. It is easily verified that this induces a one-to-one correspondence between the infinite paths in

and the infinite paths in

. We call

the nested rank of the infinite path

. Denoting by

the set of all infinite paths in

, we thus have a map

. We call

the hyperserial description of

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