Foundation Primitive Recognition Calculus Delta Probability Count Eq Card

Counting as a definition

In ordinary mathematics, a finite set has a cardinality: a natural number that says how many elements it contains. Given a property that picks out some of those elements, one can count them one by one, or one can form the subset of elements satisfying the property and then take its cardinality. These two procedures feel like the same thing, but in a formal system they need a proof.

In the Recognition Science framework, a distinction event (a property that each of a finite list of alternatives either has or fails to have) is represented as a function from the finite set of alternatives to propositions. The framework defines a count operation that filters the alternatives by the event and counts the survivors. It also defines a separate object, the event's eventFinset, which is the finite set of alternatives satisfying the event. The theorem count_eq_card states that for any such event, the count equals the cardinality of that set. The proof is immediate: the two definitions are literally the same expression, so the equality holds by definitional unfolding.

This theorem is a small but load-bearing piece of the framework's delta-native probability (probability at the most primitive finite layer). It justifies switching freely between the counting view and the set-size view when working with probabilities. The framework's headline result for this layer, delta_probability_headline, bundles the consequences: the impossible event has probability zero, the certain event has probability one, every event has probability between zero and one, inclusion of events gives monotonicity, and disjoint events have additive probabilities. All of these rest on the counting foundation that count_eq_card makes explicit.

What the theorem does not claim is anything about infinite sets, continuous probability distributions, or the physical interpretation of recognition. It only establishes a formal identity between two ways of writing the same finite count. It does not say that probabilities are objective features of the world, nor that the framework's recognition calculus is the only way to do probability. It is a definitional bridge, not an empirical or metaphysical claim.

theorem count_eq_card {N : ℕ} (E : Event N) : count E = (eventFinset E).card := by
  rfl

What this page does not claim

The theorem does not apply to infinite sets or continuous distributions. It does not assert that probabilities are objective features of the world. It does not claim that the framework's probability is the only valid notion of 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/Foundation/PrimitiveRecognitionCalculus/DeltaProbability.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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