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Inexpressibility in Exp-Minus-Log

This paper establishes that the Exp-Minus-Log system, which reduces elementary functions to a constant and a single two-place operation, can only express computable numbers, thereby proving that Chaitin's $\Omega_U$ is inexpressible within this framework.

Original authors: Mark Carney

Published 2026-05-05

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Mark Carney

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 have a magical, all-powerful kitchen. In this kitchen, you don't have a full set of tools like blenders, ovens, and mixers. Instead, you have only one special tool, let's call it the "Magic Wand" (represented in the paper as E), and a single ingredient: the number 1.

The paper by Mark Carney explores what happens if you try to cook every possible number using only this one wand and that one ingredient.

The Magic Wand's Recipe

The wand works in a very specific way. If you point it at two things, say A and B, it performs a specific trick: it takes the exponential of A (a fancy way of growing it very fast) and subtracts the logarithm of B (a way of shrinking it).

The Formula: Result = exp(A) - log(B)

The original researchers (Odrzywo lek) showed that with just this one trick and the number 1, you can actually recreate almost everything we usually think of as "standard math." You can build whole numbers, fractions, the number $\pi$ , $e$ , and even complex functions like sine and cosine. It's like being able to bake a whole wedding cake using only a single spoon and a cup of flour.

The Big Question: Can You Cook Everything?

The paper asks a deeper question: Is there anything this kitchen cannot make?

To answer this, the author compares this "Magic Wand" kitchen to a known set of numbers called EL numbers (named after mathematician Timothy Chow). The paper proves that the Magic Wand kitchen and Chow's kitchen are actually the same place. They can make the exact same list of numbers.

The "Uncookable" Ingredient: Chaitin's $\Omega$

The author then introduces a very strange, theoretical ingredient called Chaitin's $\Omega$ (Omega).

What is it? Imagine a number that represents the probability of a computer program stopping (halting) or running forever.
Why is it special? This number is "non-computable." It's not just a number we haven't calculated yet; it is mathematically impossible for any computer (no matter how powerful) to ever write down its exact digits or predict its next digit. It is a number that exists, but it is fundamentally "unreachable" by any step-by-step recipe.

The Main Discovery

The paper proves a simple but powerful rule: Every number you can make with the Magic Wand is "computable."

Here is the logic in plain English:

The Ingredients are Safe: You start with the number 1 (which is easy to calculate).
The Process is Safe: The Magic Wand only does two things: exp and log. The paper shows that if you feed these tools a number that a computer can calculate, they will spit out a result that a computer can also calculate.
The Conclusion: Since you start with a calculable number and only use calculable tools, every single number you can build in this system must be calculable.

Because Chaitin's $\Omega$ is not calculable, it is impossible to build it in this kitchen.

The Final Verdict

The paper concludes with a "Formal Inexpressibility Theorem." In simple terms:

The Magic Wand Kitchen can make a huge, infinite list of numbers (including $\pi$ , $e$ , and all the numbers used in engineering and physics).
However, there is a vast ocean of numbers that the kitchen can never touch.
Chaitin's $\Omega$ is the "poster child" for these unreachable numbers. It is a concrete proof that there are mathematical truths that cannot be expressed using this specific system of rules.

A Simple Analogy

Think of the Magic Wand system as a Lego set that only has red bricks.

You can build a house, a car, a castle, or a spaceship (these are the "computable" numbers like $\pi$ and $e$ ).
But if someone asks you to build a blue brick, you can't. No matter how many red bricks you stack, you will never create a blue one.
Chaitin's $\Omega$ is that "blue brick." It exists in the universe of math, but it is made of a material (non-computability) that simply cannot be constructed using the red bricks (the EML system) available to you.

In summary: The paper confirms that while this single-function system is incredibly powerful and covers almost all the numbers we use in daily life and science, it has a hard limit. It cannot reach the "uncomputable" numbers, proving that some mathematical concepts are forever out of reach for this specific way of writing them down.

Technical Summary: Inexpressibility in Exp-Minus-Log

Problem Statement
The paper addresses a gap in the theoretical understanding of the Exp-Minus-Log (EML) system, originally defined by Odrzywo lek [4]. This system reduces all elementary functions, constants, and operations over complex numbers to a single binary operator, $E(\alpha, \beta) = \exp(\alpha) - \log(\beta)$ , and the constant '1'. While Odrzywo lek's work was largely constructive and existential—demonstrating how to express standard elementary functions—it did not establish a formal inexpressibility theorem. Specifically, it remained an open question whether the EML system could express non-computable real numbers. This paper seeks to characterize the limits of EML expressibility by determining the computability status of all numbers representable within the system.

Methodology
The authors employ a combination of formal logic, computable analysis, and algorithmic information theory to establish the boundaries of EML. The methodology proceeds in three logical stages:

Formal Equivalence: The paper first establishes a rigorous equivalence between Odrzywo lek's EML numbers and Chow's EL numbers [2]. By providing explicit syntactic substitutions, the authors demonstrate that the field of EML numbers is identical to the field of EL numbers (the smallest subfield of $\mathbb{C}$ closed under $\exp$ and $\log$ ).
Computability Analysis: Leveraging results from computable analysis (specifically [7] and [6]), the authors analyze the computability of the operations involved. They utilize the fact that the exponential function is computable on all computable complex inputs. For the logarithm, they address the branch cut issues by proving that for any defined EML expression, the argument to the logarithm either lies on the negative real axis (where the value is computable via magnitude and $\pi$ ) or is bounded away from the branch cut by a rational distance.
Inductive Proof of Computability: Using structural induction on the syntax of EML expressions, the authors prove that every defined EML expression evaluates to a computable complex number.
Witness Selection: To demonstrate inexpressibility, the paper utilizes Chaitin's $\Omega_U$ (the halting probability of a universal prefix-free Turing machine) as a canonical witness. $\Omega_U$ is known to be left-computably enumerable but non-computable.

Key Contributions and Results
The paper yields the following primary results:

Equivalence Theorem: The set of EML-expressible numbers is proven to be exactly the set of Chow's EL numbers ($EML = EL$).
Computability Theorem: The paper proves that every EML-expressible number is computable. Specifically, for any closed EML expression $\phi$ where the value is defined, $[[\phi]]$ is a computable complex number. Consequently, the set of EML-constructable reals ( $EML_{\mathbb{R}}$ ) is a subset of the computable reals ( $\text{Comp}_{\mathbb{R}}$ ).
Inexpressibility Theorem: By combining the computability of EML numbers with the non-computability of Chaitin's $\Omega_U$ , the paper formally proves that $\Omega_U \notin EML_{\mathbb{R}}$ .
Measure-Theoretic Implication: Since the set of EML numbers is countable (due to the countability of the syntax), the set of non-EML-expressible reals has full Lebesgue measure on $\mathbb{R}$ . This includes not only $\Omega_U$ but also other classes of non-computable numbers, such as the binary expansion of the halting set, limits of Specker sequences, and Martin-Löf random reals.

Significance and Claims
The paper claims to fill a specific theoretical gap left by Odrzywo lek's original work by providing a formal inexpressibility theorem for the EML system. The significance lies in characterizing the EML system not just as a tool for generating elementary functions, but as a system strictly bounded by computability.

The authors modestly frame the result as a formal limitation: while EML is aesthetically pleasing and useful for symbolic computation and exact arithmetic (as noted in the introduction regarding CAS systems), it cannot express the "canonical" non-computable reals. The paper does not propose new applications or experimental proposals; rather, it serves as a foundational proof establishing that $EML_{\mathbb{R}} \subsetneq \mathbb{R}$ and, more specifically, $EML_{\mathbb{R}} \subseteq \text{Comp}_{\mathbb{R}}$ . The result confirms that the "closed-form" numbers expressible via this single operator are strictly limited to the realm of computable numbers.