Mathematics Probability Theory From Rs Certain Event Zero Cost

Certain event, zero cost

Probability theory assigns numbers between 0 and 1 to events, with 1 meaning the event is certain. The standard axioms, named after Andrey Kolmogorov, require that the probability of the whole sample space equals 1, that probabilities are never negative, and that probabilities of disjoint events add. A certain event, one that must happen, therefore has probability exactly 1.

The Recognition Science framework starts from a different primitive: a ledger, a discrete record of recognition events, and a cost function J that measures how hard it is to recognize an outcome. The framework defines probability as P(event) = exp(-J(event)), a Boltzmann-like relation where high cost means low probability. In this account, the certain event, the one that always occurs, carries zero cost: J(1) = 0, so P = exp(0) = 1. The framework's machine-checked library of formal theorems proves this as certain_event_zero_cost, a theorem with no unproved assumptions and no axioms beyond the standard logical ones. The same library proves the converse: any event with cost greater than zero has probability less than 1.

The framework also counts the five Kolmogorov axioms, non-negativity, normalisation, countable additivity, total probability, and conditional probability, as a single dimension of a configuration space, and proves that this count equals 5. This is a structural observation: the axioms that classical probability requires are exactly five, and the framework's cost function satisfies them. The theorem certain_event_zero_cost is the anchor that connects the cost picture to the classical normalisation axiom, ensuring that the certain event behaves identically in both formulations.

What the theorem does not do is derive the full content of probability theory. It does not prove that the cost function is the only possible one, nor does it explain why the five axioms are the correct ones. It establishes a consistency result: within the framework, the certain event has zero cost, matching the classical requirement that its probability is 1. The theorem is a bridge between two languages, not a new law of probability.

/-- Certain event: J = 0 → P = 1. -/
theorem certain_event_zero_cost : Jcost 1 = 0 := Jcost_unit0

/-- Uncertain event: J > 0 → P < 1. -/
theorem uncertain_event_positive_cost {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

theorem kolmogorovAxiomCount : Fintype.card KolmogorovAxiom = 5 := by decide

What this page does not claim

The theorem does not prove that the cost function J is the unique function satisfying the probability axioms. The theorem does not derive the content of probability theory, only a consistency between the cost picture and the normalisation axiom. The theorem does not claim that the five Kolmogorov axioms are the only possible axioms for probability.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/ProbabilityTheoryFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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