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Modifying the Field Axioms to Create Infinite and Infinitesimal Real Numbers

This paper introduces a new algebraic structure called an "ascended field" by modifying standard field axioms to allow division by zero, ultimately constructing a unique "complete s-extension" of the real numbers that preserves the totality of arithmetic operations while incorporating infinite and infinitesimal numbers.

Original authors: Brendan Santangelo

Published 2026-07-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Brendan Santangelo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you're playing with a set of magical number blocks. In the normal world of math, there's one rule you can never break: you can't divide by zero. If you try to split a pizza into zero slices, the math machine just crashes. Why? Because in standard math, if you multiply zero by anything, you always get zero. So, if someone claimed $1 \div 0 = x$ , it would mean $0 \times x = 1$ . But $0 \times x$ is always 0, never 1. The machine says "Error!" and walks away.

But what if we could tweak the rules just a tiny bit? What if multiplying by zero didn't always result in zero? What if it could result in 1, or 2, or even $\pi$ ?

That's the wild idea behind this paper. The author, Brendan Santangelo, proposes a new kind of math playground called an Ascended Field. In this playground, the rule "zero times anything is zero" is broken. Instead, zero can "touch" other numbers and change them. This allows us to finally perform the forbidden act: division by zero.

The Magic Ladder: Levels of Numbers

To make this work without everything falling apart, the author builds a new structure using "tuples" (which are just lists of numbers). But these aren't ordinary lists; they have a special property called a Level.

Think of a number's Value as its face (like the number 5) and its Level as its altitude or "zoom level."

Level 0: This is your normal, everyday number. A 5 at Level 0 is just a regular 5.
Positive Levels (1, 2, 3...): These are Infinite Numbers. A 5 at Level 1 is an "infinite 5." It's so big it's larger than any normal number you can think of.
Negative Levels (-1, -2, -3...): These are Infinitesimal Numbers. A 5 at Level -1 is a "tiny 5." It's so small it's smaller than any normal positive number, but it's not zero.

The paper proves that if you take the real numbers (the ones we use for money, distance, and time) and build this new structure, you get a Complete S-Extension. This is a unique, organized system where:

Division by zero is possible. If you divide a normal number (like 5) by zero, you don't get an error. You get a unique Infinite Number (specifically, an infinite 5). Every different number divided by zero gives you a different infinite number.
Zero is replaced by "Absolute Zero." In this new world, the old zero is gone. It's replaced by a special element called Absolute Zero (denoted as $0'$ ). This new zero acts like the old zero (adding it changes nothing, multiplying by it wipes everything out), but it allows the math to keep working even when things get weird.
Everything adds up. Unlike some other math systems that break when you try to add or subtract, this system is designed so that you can add, subtract, multiply, and divide any two numbers (as long as you aren't dividing by Absolute Zero).

The "Pixel" and the "Pixel-Size"

The author suggests a fun way to visualize this. Imagine the number line isn't a smooth, continuous line, but a digital screen made of pixels.

A normal number is a whole pixel.
An Infinite Number is like zooming out so far that a whole city fits inside one pixel.
An Infinitesimal Number is like zooming in so close that a single atom is the size of the whole screen.

The paper shows that if you multiply an Infinite Number (a zoomed-out view) by an Infinitesimal Number (a zoomed-in view) of the same "level," they cancel each other out and give you a normal, finite number back. It's like zooming out and then immediately zooming back in to the exact same spot.

What About Negative Factorials?

Here's a bonus trick the paper discovers. In normal math, you can't calculate the factorial of a negative number (like $(-5)!$ ). It's undefined. But in this new system, the author proves a formula exists:
$(-n)! = \left[ \frac{(-1)^{n-1}}{(n-1)!} \right]$
This means you can calculate factorials for negative integers, and the result is a unique Infinite Number. For example, $(-1)!$ turns out to be an infinite version of 1.

What the Paper Rules Out (and What It Doesn't)

It's important to know what this paper doesn't claim:

It does not say the old rules of math are wrong. It says the old rules are just a specific case (Level 0) of a bigger system.
It does not claim to have solved the "Division by Zero" problem for everyone yet. The paper explicitly states that while the math works, the author is still working on proving that this new system satisfies every single rule of a standard field (specifically, the rule about how addition groups things together, called "additive associativity"). The author suggests it works but admits this specific proof is coming in a future manuscript.
It does not claim these numbers exist in the physical world yet. The paper suggests that these numbers could be used to measure distances in a new way (like a "pixel" on a screen) or to understand how functions behave right at the edge of a break, but these are just ideas for future work. The paper does not say we have built a machine that uses these numbers today.

The Bottom Line

The paper introduces a new, unique mathematical structure called the Complete S-Extension. It successfully creates a world where:

You can divide by zero and get a specific, unique infinite number.
You can define factorials for negative numbers.
You have a clear hierarchy of numbers: Finite (normal), Infinite (huge), and Infinitesimal (tiny).

The author has rigorously defined how to add, subtract, multiply, and divide these numbers. While the system is mathematically consistent in most ways, the author notes that a final proof for one specific rule (associativity of addition) is still being finalized. Until then, this is a fascinating, highly structured "what-if" universe that expands the number line into a multi-level landscape of infinity and the infinitely small.

Technical Summary: Modifying the Field Axioms to Create Infinite and Infinitesimal Real Numbers

Problem Statement
The paper addresses the algebraic impossibility of division by zero within standard field theory. In a standard field, defining $1/0 = x$ leads to the contradiction $0 \cdot x = 1$ , which violates the axiom that $0 \cdot x = 0$ for all $x$ . The author posits that the obstruction to division by zero is not division itself, but the behavior of multiplication by zero. The goal is to construct an algebraic structure where multiplication by zero can yield non-zero results, thereby allowing for a defined, unique result for division by zero, while simultaneously generating a system containing infinite and infinitesimal numbers.

Methodology
The paper develops a hierarchical framework of algebraic structures, progressing from modified axioms to complex quotient spaces:

Ascended Structures and Fields: The author modifies standard field axioms to define an "Ascended Structure" ( $S$ ). Key modifications include:
- Retaining the commutative group structure for addition.
- Requiring closure under multiplication.
- Allowing $0 \cdot s \neq 0$ for some $s \in S$ .
- Replacing the standard distributive law with "Wheel Distributivity": $s \cdot (t + r) + (s \cdot 0) = (s \cdot t) + (s \cdot r)$ .
- Defining "S-Associativity" to handle associativity in the presence of non-zero zero-products.
- An "Ascended Field" further requires the existence of scalar inverses for non-zero elements in the subset $S_0 = \{x \in S \mid 0 \cdot x = 0\}$ .
Simple S-Extensions: To resolve indeterminacy in division by zero, the author defines an equivalence relation on an ascended field where $s \equiv t$ if $0 \cdot s = 0 \cdot t \neq 0$ (or if $s, t$ are scalars and $s=t$ ). The quotient space of this relation, denoted $S_0 \cup [S_0]$ , forms the "Simple S-Extension."
- In this structure, division by zero is defined: for non-zero $x$ , $x/0 = [x]$ , where $[x]$ is a unique equivalence class distinct from $x$ .
- However, arithmetic operations in the Simple S-Extension are not total; specifically, subtraction $[x] - [x]$ is indeterminate.
Tuples and Duals: To achieve totality of operations, the paper introduces the "Dual" of an ascended field ( $S^* = S \cup S^{-1}$ ), where $S^{-1}$ is an isomorphic copy of $S$ with elements assigned a "level" of $-1$ (inverse) versus $1$ (original).
- Tuples: Finite and countably infinite tuples of elements from $S^*$ are defined. Each tuple has a Level (sum of component levels) and a Value (product of component values).
- Infinite Tuples: Tuples with levels diverging to $+\infty$ or $-\infty$ are introduced. The set of $(-\infty)$ -tuples is designated as $0'$ (Absolute Zero).
Full and Complete S-Extensions:
- Full S-Extension: The set of equivalence classes of tuples $[n, x]$ (where $n$ is the level and $x$ is the value) forms the Full S-Extension. Operations are defined such that addition depends on the relative levels of operands (higher level dominates), and multiplication adds levels and multiplies values.
- Complete S-Extension: By introducing the element $0'$ (Absolute Zero, defined as $[-\infty, 0]$ ) and defining it as the additive identity, the author creates the Complete S-Extension. This structure resolves the indeterminacy of $[n, x] - [n, x]$ by defining the result as $0'$ .

Key Contributions and Results

Unique Extension: The paper proves that for any field $F$ , there exists a unique Simple S-Extension and a unique Complete S-Extension (up to isomorphism) derived from an ascended field containing $F$ as its scalar field.
Division by Zero: In the Complete S-Extension, division by zero is well-defined for all non-zero elements. Specifically, $x/0 = [x]$ (an infinite number), and $0/0$ remains indeterminate, consistent with standard algebraic intuition.
Infinite and Infinitesimal Numbers: The Complete S-Extension of the reals ( $[Z, \mathbb{R}]^*$ $[Z, R]^{*}$ ) contains:
- Finite Real Numbers: Elements $[0, x]$ corresponding to standard reals.
- Infinite Real Numbers: Elements $[n, x]$ where $n > 0$ . These are greater than (or less than) every finite real number.
- Infinitesimal Real Numbers: Elements $[n, x]$ where $n < 0$ . These are non-zero but smaller in magnitude than any positive finite real number.
- Absolute Zero ( $0'$ ): A distinct zero element that acts as the additive identity and absorbs multiplication, replacing the standard zero in the context of these extensions.
Factorials of Negative Integers: The framework allows for the definition of factorials for negative integers. The paper derives the formula $(-n)! = [(-1)^{n-1} / (n-1)!]$ , yielding unique infinite outputs for negative integer factorials.
Arithmetic Totality: The Complete S-Extension maintains the totality of basic arithmetic operations (addition, subtraction, multiplication, and division) for all elements, with the sole exception of division by Absolute Zero ( $0'$ ), which is undefined.
Ordering: A natural ordering is established for the Complete S-Extension of the reals, satisfying trichotomy. The hierarchy places infinite numbers above all finite numbers and infinitesimal numbers between zero and any finite positive number.

Significance and Claims
The paper claims to have constructed a "unique structure called a complete s-extension of the reals" that extends standard real numbers while preserving the totality of arithmetic operations. The primary significance lies in:

Resolving Division by Zero: Providing a unique, non-indeterminate result for $x/0$ (where $x \neq 0$ ) without collapsing the field structure into triviality.
Unifying Infinity and Infinitesimals: Integrating infinite and infinitesimal numbers into a single algebraic system where they are uniquely identified by a non-zero real value and an integer level.
Algebraic Consistency: Demonstrating that by relaxing the distributive property to "Wheel Distributivity" and modifying the behavior of zero, one can create a consistent algebraic system that includes these extended numbers.

The author notes that the Complete S-Extension satisfies all field requirements except for additive associativity (a claim the author states will be proven in a supplementary manuscript). The paper concludes by proposing potential applications in measuring distance via a metric function and interpreting geometric line segments as "pixels" of infinitesimal length, though these are presented as future work rather than established results.

The Magic Ladder: Levels of Numbers

The "Pixel" and the "Pixel-Size"

What About Negative Factorials?

What the Paper Rules Out (and What It Doesn't)

The Bottom Line

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