Mathematics Probability Theory From Rs Kolmogorov Axiom

Probability as a cost measure

Probability theory, as formalized by Andrey Kolmogorov in 1933, rests on a small set of axioms: probabilities are non-negative, the probability of the whole space is 1, countable additivity holds, and conditional probability behaves in a certain way. These are the rules that let a mathematician treat uncertainty with the same rigor as geometry. A machine-checked library of formal theorems, part of the Recognition Science framework, encodes these axioms as a single inductive type named KolmogorovAxiom with exactly five constructors, one for each axiom.

The framework's library proves that this type has exactly five elements, matching the classical count. It then connects the axioms to a cost function J, a recognition cost, meaning a measure of how much work it takes for reality to register an event. The connection is exponential: P(event) = exp(-J(event)), a Boltzmann-like relation. A certain event has zero cost, so its probability is exp(0) = 1. An uncertain event, one whose ratio r is not 1, has positive cost, so its probability is less than 1. These two facts are proved as theorems, not assumed.

The declaration itself is a definition, a modeling choice: it chooses to represent the five Kolmogorov axioms as one countable object and to tie them to the cost function. What the framework proves is the consistency of this interpretation: the axioms, the zero-cost certain event, and the positive-cost uncertain event all hold together in one structure called ProbabilityCert. The library's audit shows no unproved axioms and no 'sorry' gaps in these particular theorems.

This does not mean the framework has derived all of probability theory, nor that it has proved the axioms are true in the world. The axioms remain the standard starting point; the framework's contribution is a formal bridge from them to its own cost picture. A reader can now see that the five axioms fit naturally into a single object, and that the cost function's behavior at 1 and away from 1 mirrors the certain and uncertain cases exactly.

inductive KolmogorovAxiom where
  | nonNegativity | normalisation | additivity | totalProbability | conditional
  deriving DecidableEq, Repr, BEq, Fintype

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

/-- 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

def probabilityCert : ProbabilityCert where
  five_axioms := kolmogorovAxiomCount
  certain_zero := certain_event_zero_cost
  uncertain_positive := uncertain_event_positive_cost

What this page does not claim

This does not claim that the Kolmogorov axioms are derived from the cost function, only that the axioms are encoded and consistent with it. No claim is made that all of probability theory is derived within the framework. The declaration does not assert that the five axioms are the only possible ones, only that this formal object has exactly five constructors.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: