Mathematics Information Theory From Rs Shannon Axiom Count

What the axiom count proves

Claude Shannon's 1948 information theory defines entropy as a measure of uncertainty in a probability distribution. The usual formula is H = −Σ p_i log p_i, where the sum runs over possible outcomes and p_i is the probability of each. Shannon showed that any measure satisfying five natural conditions, continuity, maximality, additivity, symmetry, and subadditivity, must take this form. These five conditions are standard in textbooks and are the foundation of the field.

The Recognition Science framework models entropy as the average recognition cost of a distribution. In this account, the cost function J(x) = (x + 1/x)/2 − 1 replaces the logarithm in the entropy sum. The framework's machine-checked library of formal theorems contains a declaration, shannonAxiomCount, that proves the number of Shannon's axioms is exactly five. The proof is a direct computation: the library defines an inductive type with five constructors, one for each axiom, and then verifies that the type has exactly five elements. The result is tagged as a theorem and carries no unproved assumptions beyond the standard logical axioms of the ambient type theory.

What this theorem does not do is more important than what it does. It does not derive the five axioms from anything deeper. They are taken as given, as a definitional choice. The theorem does not prove that Shannon's entropy formula is the only possible measure of uncertainty; that uniqueness result belongs to Shannon's own theorem, which is not restated here. The library also proves that the cost function is zero when the outcome is certain (J(1) = 0) and positive when the outcome is uncertain (J(r) > 0 for r ≠ 1), which matches the intuitive requirement that entropy is nonnegative and vanishes only for a sure thing. These are separate theorems, not consequences of the axiom count.

In plain terms, shannonAxiomCount is a bookkeeping declaration. It records that the framework's formalization of information theory uses the same five axioms Shannon used. It is a small but necessary step in the larger project of showing that information theory, like other branches of mathematics, can be expressed in the recognition-cost language. A reader who wants to know whether the framework has proved Shannon's source coding theorem or the channel capacity formula will not find it in this declaration; those are separate statements, and the capacity formula appears only as a docstring comment, not as a proved theorem.

theorem shannonAxiomCount : Fintype.card ShannonAxiom = 5 := by decide

/-- Minimum entropy: J = 0 (certain outcome). -/
theorem min_entropy : Jcost 1 = 0 := Jcost_unit0

/-- Positive entropy: J > 0 (uncertain outcome). -/
theorem pos_entropy {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 five axioms are derived from the recognition framework; they are taken as given. Shannon's entropy formula is the only possible measure of uncertainty; that uniqueness is not proved here. The channel capacity formula is proved; it appears only as a comment.

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

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