E₀ = { a, b, c, d }
a.a is an action of a
a ⤳₍a.a₎ { b, c, d }
= { a₍a.a₎', b₍a.a₎', c₍a.a₎', d₍a.a₎' }
Rename:
a₍a.a₎' := a'
b₍a.a₎' := b'
c₍a.a₎' := c'
d₍a.a₎' := d'
Rebind:
a := a'
b := b'
c := c'
d := d'
Therefore:
E₁ = { a, b, c, d }
Introduction
I find that Copilot's reasoning capabilities can follow structures that resemble portions of my UnNatural Number Theory. These are not merely cardinal quantities. Rather, they are countable, non-cardinal objects whose characteristics and actions define their behavior.
The distinction about UnNatural Numbers is not that they are sets. Their information exists within an information space and may be referenced through a number's existence. An UnNatural Number may be a rock, a baseball, a door, or any other thing.
An UnNatural Number is a thing with characteristics and actions.
Planet Fragments
Consider a rock. A rock may simply be a rock, or it may be a planet fragment. The distinction arises from additional characteristics regarding its origin.
If a planet explodes and ejects fragments into space, then every surviving fragment acquires a new characteristic: its existence depends upon the destruction of the original planetary body.
Let that world be Planet ω, orbiting star θ in its final stages of stellar life. A fragment launched long ago may cross interstellar distances before eventually entering our solar system.
Information Space
Information is not necessarily contained within an object. Instead, information may be referenced through existence itself.
This concept resembles how a database record may be retrieved through a key. The object exists, while its characteristics and actions are discovered through the information space.
UnNatural Numbers
Traditional mathematics often begins with cardinality: how many objects exist.
UnNatural Number Theory instead begins with existence itself. The object precedes counting. Characteristics and actions determine the meaning of a number.
Closing Thoughts
Arithmetic is only one operation available to intelligent systems. Humans and computers are also capable of reasoning about trajectories, physical objects, information systems, astronomy, finance, and abstract concepts.
The purpose of UnNatural Number Theory is to explore what occurs when numbers are liberated from cardinality and are instead permitted to represent things.
JonesOnie FX