Existential Geometry: Omega (w) in Mathematics

The thesis of this paper is to ground advanced mathematical concepts within the objective bounds of existent reality. Infinity provides a valuable tool but one that requires further bounding. Where those boundaries lie cannot be subjective to be useful. Every system is bounded by its final state, due to the unitary direction of entropy. Without entropy each state would be equally considered a bound. However, entropy dictates that only the beginning and final state is inviolable where all states in between are bound by those states. Within this, there can still be subjective boundaries placed that may be useful for different purposes, but the boundaries of a system cannot both be exceeded and considered valuable. Any value not attainable in our current reality should be considered infinite. This places a lower bound on infinity within a system, where finite stops.

Disclaimer

I am not an academic and have not before attempted to formulate ideas as complex as this nor as rigorously as I attempt to do so here. Please excuse the amateur formatting and focus on the underlying thoughts, because it absolutely is formatted by an amateur.

Definitions

Ρ: (big rho) Represents the concept of resources and potential.

ρ: (little rho) Represents a real number of resources and potential.

Φ: (big phi) Represents the overall fragility of resources; all resources have fragility as defined by their internal entropy.

φ: (little phi) Represents the amount of fragility within a resource or system. Fragility of zero (0) would imply indestructibility and one (1) would imply immediate dissolution.

Node: Resource dedicated to supporting the system

Edge: Resource dedicated to transaction of resources between Nodes

Transaction: Any allocation of resource from one node to another along an edge

System: A set of nodes connected by edges along with free resources are transacted. Note since a system is entirely composed of resources it is also a resource in aggregate.

Actor: A special system that is capable of transacting with and therefore shaping the system in which it resides.

ω: (little omega) Represents the amount of entropy within a system.

Ω: (big omega) Represents the maximum entropic state of a system.

Ω Planes: Dimensional planes bounded by Ω values of a system, not infinity.

Ω Horizon: The boundary beyond which numbers lose meaning, based on the ranges available to the Ω Plane of a system. Beyond this point can be considered “infinity.”

ω_ρ (little omega sub rho) Represents the resources that could be used, or the entropy that could be released by an actor, object, or actor using an object as opposed to Ω, which is the maximum amount of entropy a system can sustain.

Axioms

 

The Axiom of Arbitrariness

 ρ_m, ρ_n; ρ_m ≡ ρ_n → ρ_m ≠ ρ_n 

While there may be a high degree of equivalence among resources, no two resources are the same, and there is no such thing as arbitrariness. Even if the only reason a specific resource is chosen over another is “time and place” that is not arbitrary. It is the outcome of the entire system in which the resource resides up until that point. While an equivalent resource may be available it is not arbitrary which is chosen.

The Greater Potential Axiom

ω_ρ(S) > Ω(S) 

Entropy potential is greater than the overall possible entropic state, for all systems. This allows for the concentration of potential entropy in any given region to exceed Ω and allow for instantaneous heat death of a system upon release.

Consider a house of cards. It is a system because it can be defined as a system. It is well structured since there are only so many ways to stack cards to form a house of cards, yet it is fragile. The act of removing a bottom row card is inevitably going to cause the collapse of the entire system. Each card in the bottom row has this equal potential to cause heat death of the system (all cards scattered on the floor). This concept represents the combined potential for entropy within a system, therefore the potential entropy in the system is greater than the total capacity for entropy in the system.

The Resource Axiom

Ρ  ≡ M ≡ E ≡ I ≡ G 

Mass, Energy, Information, and Gravity are all equivalent within Ω calculations and are grouped under the greater category of Resource.

This is grounded in E=MC² and the physical nature of reality. This is not a simplification but a fact of heat death. More understandably this highlights the one-use-only of nature. You can either build a house with wood, burn it for warmth, pulp it for paper and draft books with it, or build a house you then pulp for books you then later burn for warmth. You cannot build with it, create information from it, and burn it at the same time. And once burned it is ashes that can never become the same original pieces of wood again. Heat Death, or Big-O Omega, represents the ashes. It is not germane whether the ashes were ever a house. Nor can that be determined from this end-state; hence all are equivalent.

Gravity is a consequence of concentration of resource and its interaction with the fabric of space-time by curving it. This concentration of resource itself creates resource, as becomes obvious in the “house of cards” anecdote when considering what resource holds the cards together. However – in the heat death state of a system there can be no concentration of resource capable of producing gravity, therefore while a consequence of other phenomena it is equivalent in its definition as a useable resource that experiences entropy, since the resources creating the concentration are themselves subject to entropy as are resources under gravity’s influence.

* Words on a page may represent a primitive form of information storage but it cannot be said that it is not information storage.

The Boundary Axiom

  Ω(S) <   ω_ρ(S) < ∞   

No system is infinite; to be defined a system, it must be finite with defined bounds. Those bounds are a property of the system and not arbitrary.

This places a lower bound on infinity for a given system, including the universe itself. It does not, however, claim to make any determinations about whether the universe is itself an open system or a closed system. However, for it to be open it must exist within a larger system, and the concept of Omega still applies.

This axiom states that a boundary must exist for a system to be defined and entropy is the defining factor for that boundary.

The Reality Axiom

 0 < ω ≤ Ω < ω_ρ < ∞ 

Omega, and by extension Omega Potential, are inherently grounded by reality and are therefore quantifiable. The techniques to quantify them may not currently exist but that does not betray their theoretical quantifiability. Any value greater than the entropic value of a system should be considered “infinity” within the frame of that system.

This axiom states that since boundaries must exist, as per the boundary axiom, they must be real and definable. Any boundary outside the system is superseded by the boundary of that system from within the point of view of that system. While there may be something outside of the universe, unless we can see outside of it that fact is moot to those of us within it. Any boundary placed outside of the universe from within it is moot and meaningless, and therefore not real; that is infinity.

The Axiom of Transactions

T(ρ) = Δρ - ερ

A transaction is a change in resources. The magnitude of resource exchange for a system due to a transaction can be zero in the case of an equal exchange transaction, or indeed for all transactions taking place within a system. However, the efficiency of that transaction (epsilon), which can never be 100%, requires that a non-zero number of resources be lost to entropy, and the magnitude lost is directly proportional to the magnitude of the resources transacted.

A transaction is any allocation of resources within a system or with another system. All transactions have a direction and a resource. The magnitude of resources and the rate of the transaction cause entropic friction. No rate is slow enough to avoid this. No resource small enough. If it has resource value and is moving, it is causing friction. This is due to the second law of thermodynamics and the quantum fluctuations of vacuum – even in space this “devil’s tax” is paid. This implies that even in a perfect vacuum, given sufficient distance, any object in motion would be brought to rest by these fluctuations without its own source of motion.

The System Resource Axiom

  Ρ(S) = ρ_system + ρ_free  

In any real-world system, it is not sufficient for there to be the concept of nodes and edges from Graph Theory, but those nodes and edges must exist. That means they must consume a portion of a system’s resources to function. Edges tend to be considered liminal – however the strength of an edge is never 100% and cannot be abstracted. A bridge can collapse. A vessel can run out of fuel on a voyage. Edges are composed of resources every bit as much as Node. Both are resources consumed by the system and not by entities within the system.

All edges and nodes are composed of resources. Not all resources are edges and nodes. This is true for any system that is not purely logical. For an edge to exist it must be created. Even if that creation is wandering through the woods from basecamp to the summit of a mountain for the first time to create a trail. Nothing that matters can be considered not to exist. Liminal spaces matter because we need them to transact resources between nodes.

Due to the axiom of arbitrariness, there is always a distinction between a “system” resource and a “free” resource even if the resources are otherwise equivalent. Every resource occupies a region in spacetime, and this is the result of the state of the system at any given point in time.

This is a work in progress and still in the first principles phases. I welcome feedback but know that whether I choose to listen or not is not a judgement of the advice but an acknowledgement of the fragility of these initial stages. Should any of this work prove valuable, it will inevitably be taken out of my hands as I cannot own these ideas; however I am currewntly their steward.