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Bernoulli Void

The void type, denoted by ∅ or Void, is the empty type with no values. This document explores the theoretical properties of B_Void.

Theoretical Properties

Void Type Characteristics: - Cardinality: |Void| = 0 - Values: None (cannot construct values of this type) - No functions out of void except the vacuous function

Bernoulli Void Analysis

Confusion Matrix: For B_Void, we have a 0×0 confusion matrix:

(empty matrix)

Fundamental Measures: - Order: 0 (no entries, no parameters) - Rank: 0 (empty matrix has rank 0)
- Entropy: H = 0 (no values to be uncertain about)

Why B_Void = Void

Since there are no values of the void type, there's nothing to approximate:

No Values to Observe: - Can't construct Void values to begin with - Can't construct B_Void observations - The approximation question is vacuous

Vacuous Truth: Any statement about B_Void approximation is vacuously true since there are no instances to verify: 

∀x ∈ Void: Pr{B_Void(x) ≠ x} = ε  // vacuously true for any ε

Connection to Other Types

Compositional Role: The void type plays crucial theoretical roles:

  1. Sum types: B_{Void + X} ≅ B_X (void contributes nothing)
  2. Product types: B_{Void × X} ≅ B_Void (void annihilates everything)
  3. Function types:
  4. B_{Void → X} has cardinality 1 (unique vacuous function)
  5. B_{X → Void} represents partial functions (possibly undefined)

Algebraic Zero: The void type serves as the additive identity and multiplicative zero: - X + Void ≅ X (additive identity) - X × Void ≅ Void (multiplicative zero) - B_X + B_Void ≅ B_X - B_X × B_Void ≅ B_Void

Practical Significance

Error Handling: In implementations, B_Void represents: - Impossible cases in pattern matching - Error states that shouldn't occur - Compiler-enforced impossibility

Type System Completeness: B_Void ensures the Bernoulli type system is: - Algebraically complete (all type constructors defined) - Compositionally sound (operations compose consistently) - Theoretically elegant (no special cases needed)

Information Theory

Degenerate Information: - No information content (can't observe anything) - No uncertainty (no values to be uncertain about)
- Vacuous inference (all inference questions are meaningless)

The void type represents the ultimate degenerate case where approximation, observation, and inference all become vacuous concepts due to the absence of any values to approximate.