Mathematics Probability Theory From Rs Kolmogorov Axiom Count

The five axioms

Probability theory, as taught in every statistics course, rests on a small set of axioms. The most common formulation, due to Andrey Kolmogorov in 1933, uses five: non-negativity, normalization, countable additivity, total probability, and conditional probability. These five rules govern how probabilities behave: they cannot be negative, the probability of the whole sample space is 1, probabilities of disjoint events add, the total probability over a partition sums to 1, and conditional probability obeys the product rule. The five-axiom structure is standard; the count itself is not in dispute among mathematicians.

The Recognition Science framework formalizes this count in its machine-checked library of formal theorems. The declaration kolmogorovAxiomCount establishes, as a proved theorem, that the number of Kolmogorov axioms is exactly 5. The proof is a direct computation: the framework defines an inductive type with five constructors, one for each axiom, and then verifies by decision that the cardinality of this type equals 5. The theorem carries no assumptions and no axioms beyond the ambient logic; the library reports zero sorry and zero axiom.

In Recognition Science, the number 5 is not arbitrary. The framework models probability as a cost measure: P(event) = exp(-J(event)), where J is the recognition cost. A certain event has zero cost, so P = exp(0) = 1; an uncertain event has positive cost, so P < 1. The framework proves both facts as theorems. The five axioms align with the framework's configurational dimension D = 5, a coincidence the framework treats as structural rather than accidental. The library bundles these results into a structure called ProbabilityCert, which packages the axiom count, the certain-event zero-cost theorem, and the uncertain-event positive-cost theorem into one certificate.

What the declaration does not claim is important. It proves a count, not the content of the axioms. It does not derive the axioms from first principles, nor does it show that probability must take this form. The theorem merely states that the standard five axioms number five. It also does not claim that the framework's cost-based interpretation of probability is the only one, or that it replaces the classical measure-theoretic foundation. The certificate assembles the results, but the interpretive step, equating probability with exp(-J), is a modeling choice, not a proved consequence.

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

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

What this page does not claim

The declaration does not derive the axioms from more basic principles. The equality P = exp(-J) is a modeling choice, not a proved consequence of the axiom count. The theorem does not prove that the five-axiom formulation is unique or superior to other axiomatizations.

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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