Ordered Field of Infinite Numbers

• Ordered Field of Infinite Numbers is a totally ordered field that extends the real numbers by incorporating both infinitesimal and infinite elements.
• Constructions like the surreal numbers, Euclidean numbers, and smooth-sequence fields use recursive definitions, transfinite sums, and asymptotic limits to formalize diverse orders of magnitude.
• These fields establish a universal framework linking classical analysis to nonstandard methods and transseries, offering robust analytic and algebraic properties in extended number systems.

An ordered field of infinite numbers is a totally ordered field extending the real numbers, containing elements that are infinitesimal (smaller than any positive real) and infinite (exceeding any real), and in many constructions, encapsulates an entire hierarchy of such magnitudes. These structures are crucial to modern non-Archimedean analysis, provide universal domains for asymptotic growth phenomena, and possess canonical connections to both classical real analysis and transseries/hyperseries formalisms. The definitive constructions are the surreal numbers $\mathbf{No}$, hyperreal fields (such as the Euclidean numbers $E$), and fields of smooth-sequence limits such as $\mathbb{I}_n$.

1. Construction of Ordered Fields with Infinite and Infinitesimal Elements

The prototypical example is the field of surreal numbers $\mathbf{No}$, introduced by J. H. Conway. Every surreal number is built recursively from pairs of sets (the "birthday form")

$x = \{\, x_L \mid x_R \}$

where $x_L$, $x_R$ consist of previously constructed numbers, subject to the constraint that no element of $x_L$ is $\ge$ any element of $x_R$. Operations $E$0, $E$1, $E$2, and multiplicative inverse are recursively defined, turning $E$3 into a proper class real-closed, totally ordered field containing $E$4, $E$5, and the class of all ordinals as substructures (Bagayoko et al., 2023).

Several set-theoretic subfields exist with similar properties. For example, the Euclidean numbers $E$6 are a saturated non-Archimedean real-closed field constructed via transfinite sums indexed by all ordinals below the first inaccessible cardinal $E$7. Every $E$8 can be uniquely expressed as a transfinite sum of real numbers

$E$9

where all but finitely many $\mathbb{I}_n$0 vanish. The axioms governing these sums ensure $\mathbb{I}_n$1 is closed under field operations, has a compatible ordering, supports a topology based on partial sums, and contains a copy of the real numbers as well as canonical embeddings of ordinal arithmetic (Benci et al., 2017).

Another approach is via smooth sequences and asymptotic expansions: the field $\mathbb{I}_n$2 is defined as

$\mathbb{I}_n$3

where the "leading term limit" $\mathbb{I}_n$4 captures polynomial/exponential order growth of non-oscillating real sequences. The field operations and order are defined combinatorially on such asymptotic expansions (Paterson, 2011).

2. Algebraic and Order-Theoretic Structure

All major constructions yield real-closed fields that are non-Archimedean—that is, they admit infinitesimals $\mathbb{I}_n$5 with $\mathbb{I}_n$6 for every standard $\mathbb{I}_n$7, as well as elements $\mathbb{I}_n$8 such that $\mathbb{I}_n$9 for all standard $\mathbf{No}$0. In every case, the field order extends that of $\mathbf{No}$1 and interacts compatibly with addition and multiplication.

For example, in $\mathbf{No}$2, the total order is defined on representatives $\mathbf{No}$3 by $\mathbf{No}$4 if $\mathbf{No}$5. Every field axiom holds, including the existence of multiplicative inverses (for nonzero elements), and the order is total and compatible with all operations (Paterson, 2011).

The fields constructed from the surreals, especially subfields $\mathbf{No}$6 and their completions $\mathbf{No}$7 (with $\mathbf{No}$8 a "day" $\mathbf{No}$9 in Conway's chronology), retain the property that every positive $x = \{\, x_L \mid x_R \}$0 are related via a $x = \{\, x_L \mid x_R \}$1-Archimedean principle: there exists an ordinal $x = \{\, x_L \mid x_R \}$2 such that $x = \{\, x_L \mid x_R \}$3 (Lisica, 2024):

Field	Infinitesimal/Infinite	Real-Closed	Archimedean Property
$x = \{\, x_L \mid x_R \}$4	Yes	Yes	Non-Archimedean
$x = \{\, x_L \mid x_R \}$5	Yes	Yes	Saturated, non-Archimedean
$x = \{\, x_L \mid x_R \}$6	Yes	Yes	$x = \{\, x_L \mid x_R \}$7-Archimedean Property
$x = \{\, x_L \mid x_R \}$8	Proper class	Yes	Strongest non-Archimedean

3. Analytic and Topological Aspects

Topologies can be naturally introduced on ordered fields of infinite numbers, extending the order topology of $x = \{\, x_L \mid x_R \}$9. For $x_L$0, the topology is given by convergence of partial transfinite sums $x_L$1 as $x_L$2 in a filter topology; in $x_L$3, the order-topology is finer, and linear order intervals form a basis, mirroring familiar constructions from real analysis but extending to handle transfinite Cauchy sequences and completions (Benci et al., 2017, Lisica, 2024).

Associated with each completion $x_L$4 is a robust fragment of real analysis: every positive $x_L$5 has a unique $x_L$6th root in $x_L$7 for each $x_L$8, and every odd-degree polynomial over $x_L$9 admits a root in $x_R$0 (with the algebraic closure $x_R$1 being algebraically closed) (Lisica, 2024). This extends classical completeness and algebraic properties of $x_R$2 to these new transfinite domains.

4. Relationship with Transseries and Hyperseries

The theory of transseries and hyperseries provides a formal calculus for comparing asymptotic growth rates and defining operations such as exponentials and logarithms over infinitely large and small domains. In this setting, every surreal number can be uniquely represented as the value at $x_R$3 of a hyperseries—a formal sum with a well-founded transfinite tree structure allowing iterated exponentiation and logarithms of arbitrary ordinal depth (Bagayoko et al., 2023).

The canonical evaluation map $x_R$4 identifies the proper class field of hyperseries $x_R$5 with $x_R$6, establishing that monomials in $x_R$7 correspond precisely to infinite monomials in $x_R$8, and that the natural orderings, sum, and product operations are preserved under this bijection. This formalizes $x_R$9 as the unique universal ordered field of all "growth rates" (Bagayoko et al., 2023).

5. Universal Properties and Embeddings

A central result is that $x_L$0 is universal among ordered fields with infinitesimal and infinite elements. Every divisible ordered abelian group $x_L$1 and every real-closed field $x_L$2 can be s-hierarchically and initially embedded as an initial subtree and initial subfield of $x_L$3. More concretely, any truncation-closed, cross-sectional subfield of a Hahn field with suitable structural properties (e.g., log-Hahn fields, transserial fields) can be embedded as an initial subfield of $x_L$4 (Ehrlich et al., 2020). The image of classical transseries fields—such as the field of logarithmic-exponential transseries—under canonical mappings is always an initial subfield of $x_L$5.

In the Euclidean numbers construction, the field $x_L$6 is the unique saturated real-closed hyperreal field of cardinality $x_L$7, containing both $x_L$8 and the semiring of numerosities for labeled sets, and every ordered field of cardinality $x_L$9 embeds into $\ge$0 (Benci et al., 2017).

6. Categorical Hierarchies and Completion Towers

The approach in (Lisica, 2024) and (Bagayoko et al., 2023) constructs an ascending tower of topological fields

$\ge$1

with each $\ge$2 and $\ge$3 containing "larger" infinities and "smaller" infinitesimals, indexed by increasingly complex ordinals such as $\ge$4 ($\ge$5 times). Each completion is realized as the field of limits of Cauchy sequences of length $\ge$6 in the order topology, and each possesses the analytic and algebraic closure properties extending those of the previous field (Lisica, 2024).

7. Summary and Canonical Role

Ordered fields of infinite numbers, as instantiated by the surreals, the Euclidean numbers, and asymptotic fields like $\ge$7, provide the foundation for a unified treatment of infinite and infinitesimal analysis, generalized measure theory, and the formalization of asymptotic growth rates. The field $\ge$8, together with its initial subfields and completions, serves as a universal domain for real-closed ordered fields, exponential extensions, and the algebraic representation of all conceivable rates of infinity and infinitesimality, subsuming both classical and nonstandard analysis (Bagayoko et al., 2023, Ehrlich et al., 2020, Lisica, 2024, Paterson, 2011, Benci et al., 2017).