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Are there infinities that are “larger” than others? If so, how to carry out computations with infinite quantities, like , , , ? The mathematical study of this kind of questions started during the end of the 19^{th} century.
On the one hand, Cantor introduced ordinal and cardinal arithmetic [6, 13], which allowed him to quantify the “size” of an infinite set. Slightly anterior to Cantor's work, but less well known, du Bois-Reymond [9, 10, 11] developed a “calculus of infinities” to deal with growth rates of real functions in one variable, representing their “potential infinity” by an “actual infinite” quantity.
At first sight, Cantor's discrete infinities (generalizing natural numbers) and du Bois Reymond's growth orders (generalizing real numbers) are of a very different nature. We will survey the subsequent developments of these theories and recent progress towards their ultimate unification [2].
Note: Joint work with Matthias
Occasion: ALGOS 2020: ALgebras, Graphs and Ordered Sets, Nancy, August 27
Dedicated to: 75^{th} birthday of Maurice
Documents: slideshow, TeXmacs source
Extended abstract |
Ordinal numbers can be regarded as a generalization of natural numbers, where we are “allowed to count beyond all numbers that we already constructed”:
In Conway's theory of surreal numbers [7, 20], we may also construct numbers between already known numbers: given two sets of surreal numbers, there exists a simplest surreal number with . This theory naturally extends Cantor's theory of ordinal numbers:
More interestingly, arithmetic operations on surreal numbers can be defined in a surprisingly elegant way, after which the class of all surreal numbers turns out to be a totally ordered real closed field that contains . For instance:
An interesting question is which other real calculus operations “naturally” extend to the surreal numbers. For instance, Gonshor defined an exponential on with the same first order properties as the usual exponential [14]. More recently, Berarducci and Mantova showed how to define a derivation with respect to on [4].
Du Bois-Reymond's ideas were put on a firm bases by Hausdorff [17] and Hardy [15, 16]. Hardy introduced the set of “logarithmic-exponential functions” such as
He proved the remarkable fact that the set of germs of such functions at infinity form a real closed differential field. This was later generalized by Bourbaki [5], who defined a Hardy field to be any field of germs at infinity that is stable under differentiation.
Another formal direction of generalization is to consider so-called “transseries”, which are infinite logarithmic-exponential expressions such as
Transseries were introduced independently by Dahn–Göring [8] and Écalle [12], and their theory was further developed in [18, 19, 1]. Again, it turns out that the class of all transseries forms a totally ordered differential field.
We have now seen three types of real closed differential fields with infinitely large quantities: the surreal numbers, Hardy fields, and the field of transseries. In each of the three cases, it turns out that the derivation and the ordering satisfy additional compatibility properties like . The notion of an “H-field” captures the most obvious common first order properties of this kind.
The field of transseries also satisfies several less obvious first order properties such as the intermediate value theorem [19]: given a differential polynomial and in with , there exists a with . An H-field is said to be “H-closed” if it satisfies this and a few other closure properties. The main result of [1] is that the elementary theory of is completely axiomatized by the axioms of H-closed H-fields. Moreover, we proved a quantifier elimination theorem for a natural expansion of this theory.
The language of H-fields allows us to make the relations between surreal numbers, Hardy fields, and transseries more precise. For instance, the ordered differential field of surreal numbers is elementary equivalent to [3]. We conjecture that the same holds for all maximal Hardy fields. We also conjecture that there exists a natural isomorphism between and a suitable field of “hyperseries”—a generalization of transseries with functions such as the solution of . We refer to [2] for detailed statements and partial results.
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