Universal Number Set (UNS)

A redefinition of what numbers can be — and what they can model.

Universal Number Set (UNS) Runtime Playground

What Is UNS?

The Universal Number Set (UNS) is an extended number system built around a simple but transformative idea:

A number is not a single value. A number is a distribution of values across a universe of microstates.

In UNS, every “number” is actually a function from microstates to values, defined on a normalized measure space.1 This design gives UNS mathematical behaviors that classical numbers cannot express — without breaking compatibility with real and complex numbers.

Formally:

[ U = (X, \mu), \quad \mu(X) = 1 ]

[ u : X \to \mathbb{C}, \quad \psi : X \to \mathbb{C}, \quad \int |\psi(x)|^2 \, d\mu = 1 ]

[ \operatorname{read}(u \mid \psi) = \int u(x) \cdot |\psi(x)|^2 \, d\mu ]

Classical numbers embed as constant functions (e.g., const(7)(x) = 7), so UNS strictly contains ℝ and ℂ while extending them.

UNS is both:

If classical numbers are points, UNS numbers are landscapes.

Why UNS Was Created

The origin of UNS emerged from exploring a series of conceptual questions:

1. Why do classical numbers fail in certain contexts?

Classical math forces numbers to be:

But many real and conceptual systems are:

Classical numbers weren’t designed with this worldview in mind.

2. Can we create a number system that tolerates contextuality?

UNS introduces:

This allows UNS numbers to behave differently depending on the state through which they are viewed—yet still follow stable, formal rules.

3. What if dimensionality were not intrinsic?

One of UNS’s most striking properties:

A UNS number can appear point-like or N-dimensional depending on the state, yet yield the exact same classical readout.

This mirrors deep symmetry principles found in physics and information theory, echoing Hilbert-space equivalences while remaining agnostic to a single inner-product structure.2

UNS formalizes this by design.

4. Can we handle undefined classical operations without breaking the system?

When classical math encounters singularities (like division by zero), UNS instead generates:

novel values — formally typed, traceable, valid outputs that extend the number system without contradiction.

This makes UNS an open number universe: closed under all lifted operations, including those that classically fail.

What UNS Is Good For

UNS is not intended to replace classical mathematics. Instead, it is ideal for modeling or experimenting with systems characterized by:

Potential applications include:

UNS combines formal mathematical rigor with an exploratory design ethos.

Key Documents in This Repository

📘 Universal Number Set — RFC Specification

The formal definition of UNS: grammar, rules, operators, semantics, and foundational axioms. → RFC/UNS_RFC.md

📗 Operator Extensions

Defines extended operators such as cancellation, along with any additional lifted functions or helper constructs. → Runtime/Specification/UNS_Runtime32_Spec.md#11-uns-operator-extensions

📄 .unse — Universal Number Set Expression Files

UNS includes a dedicated file type for storing UNS programs and expressions. The .unse spec describes:

🧭 Guided Discovery Document

A narrative, intuitive walkthrough of the thinking behind UNS: microstates, distributions, dimensional equivalence, novel values, lifted operators, and more. This is not part of the spec but is ideal for understanding the why behind the system. → Included in repo as UNS_Guided_Discovery.md

🖼 Logo & Symbol Assets

A stylized UNS symbol (derived from the letters U–N–S) and banner graphics suitable for web, documentation, and packaging. → Assets are being curated; see TRADEMARKS.md for current usage guidance until the vector set lands in-repo.

🧪 Examples & Reference Expressions

Sample .unse files showing:

→ Examples/

How to Use UNS

You can use UNS in two main ways:

1. As a Language

Write and evaluate .unse expressions describing UNS values and operations.

2. As a Conceptual Framework

Use UNS ideas to:

Project Philosophy

UNS is built around a few core principles:

🟦 Structural clarity

Every UNS number has an internal structure you can inspect.

🟩 Extensibility without chaos

New operators and new values can be added without breaking old ones.

🟧 Mathematical honesty

No contradictions allowed; anything consistent is permitted.

🟪 Interpretational neutrality

UNS does not force an interpretation—classical numbers, quantum-like views, probabilistic views, or purely abstract perspectives all fit.

Getting Started

  1. Read the RFC Specification for the formal rules.
  2. Browse the Guided Discovery for intuition and background.
  3. Explore the example .unse files.
  4. Use the designer-instruction prompt if integrating UNS into tooling or automated systems.

License

Parts of this repository are open source; others are closed or proprietary and not redistributable. Each directory contains its own license file—please review before use. For more info, see the repository’s license files for all applicable terms. Licensing Overview.md.

Errors and Contradictions

There may be errors, contradictions or inconsistencies across the various documents. I’ve tried to prevent such things, and correct them when found, but if you spot anything questionable, please open a bug report.

The most accurate examples of valid expressions should be found in the Wiki from Section 09 onward.

Contributing

UNS is a research-aligned, exploratory project. Contributions are welcome in the form of:

Please open an issue or submit a PR.

  1. A. N. Kolmogorov and S. V. Fomin, Measure, Lebesgue Integrals, and Hilbert Space, Academic Press, 1957. ↩

  2. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955. ↩