[Paper Review] On the infinity of infinities of orders of the infinitely large and infinitely small

Leonhard Euler, Jordan Bell|ArXiv.org|May 14, 2009

Mathematical Dynamics and Fractals20 citations

TL;DR

Euler investigates the existence of infinitely many orders of infinity and infinitesimals, demonstrating that between any two infinite or infinitesimal quantities, no matter how far apart, there exist infinitely many intermediate orders. Using geometric progressions and higher transcendental functions—particularly logarithms and exponentials—he shows that new, distinct orders of infinity and infinitesimally small quantities emerge beyond any finite hierarchy, proving that the spectrum of infinities is not only infinite but unbounded in complexity.

ABSTRACT

Translation (by J.B.) from the original Latin of Euler's "De infinities infinitis gradibus tam infinite magnorum quam infinite parvorum" (1780). E507 in the Eneström index. Euler discusses orders of infinity in this paper. In other words this paper is about how different functions approach infinity or 0 at different rates. I was not certain about what Euler means by "infinities infiniti". Probably he means that $x,x^2,x^3$, etc. are infinitely many orders of infinity, and $\log x,(\log x)^2, (\log x)^3$, etc. are infinitely many orders of infinity, so the combinations of them are an infinity of infinities of orders of infinity. In fact Euler mentions other orders of infinity in this paper. It would be worthwhile to study this paper more to figure out exactly what Euler means here. Another translation of the title is "On the infinitely infinite orders of the infinitely large and infinitely small". Here's another place Euler uses the phrase "infinities infiniti". The phrase "infinities infiniti" from the title is used by Euler also in section 21 of E302, "De motu vibratio tympanorum". Truesdell translates this phrase on p. 333 of "The rational mechanics of flexible or elastic bodies" as "infinity of infinities". I'd like to thank Martin Mattmueller for clearing up some questions.

Motivation & Objective

• To establish that the concept of infinity is not a single entity but comprises infinitely many distinct orders of magnitude.
• To demonstrate that between any two infinite or infinitesimal quantities, even in a geometric progression, there exist infinitely many intermediate orders.
• To show that higher analysis—particularly logarithmic and exponential functions—produces new orders of infinity and infinitesimals not captured by polynomial or power-based hierarchies.
• To rigorously analyze the asymptotic behavior of integrals involving transcendental functions in the context of infinitesimals and infinities.
• To resolve the conceptual gap in existing mathematics regarding the unbounded nature of orders of infinity, especially in the context of calculus and analysis.

Proposed method

• Uses geometric progressions of the form $1, x, x^2, x^3, \ldots$ where $x$ is an infinite quantity, to show that each term is infinitely greater than the prior and infinitely smaller than the next.
• Applies recursive substitution: if $x = y^{1000}$, then $y = \sqrt[1000]{x}$ is still infinite, allowing for another 1000 intermediate orders between 1 and $y^{1000} = x$, thus generating a new hierarchy.
• Introduces logarithmic and exponential functions as sources of new orders: for infinite $x$, $\log x$ is infinite but infinitely smaller than any $x^{1/n}$, no matter how large $n$.
• Analyzes integrals of the form $\int \frac{x^n}{e^{\alpha/x^\beta}} dx$ for infinitesimal $x$, showing they yield closed-form results like $\frac{1}{\alpha\beta} \frac{x^{n+\beta+1}}{e^{\alpha/x^\beta}}$, which represent new orders of infinitesimals.
• Uses logarithmic differentiation to derive differential forms for functions involving $\log(1/x)$ and $e^{\alpha/x^\beta}$, enabling integration of otherwise non-elementary expressions.
• Demonstrates that even when higher-order terms like $x^m \log^n x / e^{\alpha/x^\beta}$ are introduced, the integral remains expressible as $\frac{1}{\alpha\beta} \frac{x^{k+\beta+1} u^m}{v}$, revealing a consistent framework for new orders.

Experimental results

Research questions

• RQ1 Can there be infinitely many distinct orders of infinity between 1 and an infinite quantity $x$?
• RQ2 Do logarithmic and exponential functions generate new orders of infinity or infinitesimals not representable within standard power-based hierarchies?
• RQ3 Can integrals involving transcendental functions like $e^{\alpha/x^\beta}$ yield new, distinct orders of infinitesimals beyond polynomial or rational forms?
• RQ4 Is it possible to construct a hierarchy of infinitesimals such that each is infinitely smaller than the previous but infinitely larger than the next?
• RQ5 Are there analytical expressions for areas under curves defined by $y = x^n e^{-\alpha/x^\beta}$ that reveal new orders of infinitesimals?

Key findings

• Between any two terms in the geometric progression $1, x, x^2, \ldots$ where $x$ is infinite, there exist infinitely many intermediate orders of infinity, even when the exponent difference is fixed at 1000.
• If $x = y^{1000}$, then $y = \sqrt[1000]{x}$ is still infinite, allowing the construction of another 1000 intermediate orders between 1 and $x$, and this process can be iterated indefinitely.
• For any infinite $x$, $\log x$ is infinite but infinitely smaller than $x^{1/n}$ for any finite $n$, proving that logarithmic growth defines a distinct, lower order of infinity.
• The integral $\int \frac{x^n}{e^{\alpha/x^\beta}} dx = \frac{1}{\alpha\beta} \frac{x^{n+\beta+1}}{e^{\alpha/x^\beta}}$ holds for infinitesimal $x$, showing that such expressions define new, distinct orders of infinitesimals.
• When $y = \frac{a x^n}{e^{\alpha/x^\beta}}$, the area under the curve is $\int y dx = \frac{1}{\alpha\beta} x^{\beta+1} y$, which is a closed-form expression revealing a new class of infinitesimal quantities.
• Even when logarithmic terms $\log(1/x)$ are included, the integral $\int \frac{x^k u^m}{e^{\alpha/x^\beta}} dx$ remains expressible as $\frac{1}{\alpha\beta} \frac{x^{k+\beta+1} u^m}{e^{\alpha/x^\beta}}$, confirming the existence of a consistent, unbounded hierarchy of infinitesimals.