Mathematics Set Theory From Rs Power Set Q3 Eq 256

The power set of Q₃

In classical set theory, the power set of a set S is the collection of all subsets of S. If S has n elements, its power set has 2^n elements, because each element either belongs to a given subset or it does not. For a set with three elements, the power set has 2^3 = 8 elements. The power set of that power set, the collection of all subsets of an 8-element set, has 2^8 = 256 elements. This is the elementary counting fact at the center of the declaration.

The Recognition Science framework's machine-checked library of formal theorems records this fact as a named theorem, powerSetQ3_eq_256. The theorem states that the number 2^8, which the framework defines as the power set of Q₃, equals 256. Here Q₃ is the framework's name for a set with three elements, and the framework models the power set construction as the ordinary mathematical operation. The proof is a direct computation: the definition and the equality are checked by the machine, with no gaps and no additional assumptions.

The framework also links this counting fact to a choice of five Zermelo-Fraenkel axioms: extensionality, pairing, union, power set, and infinity. These five are among the nine standard ZF axioms, and the framework selects them as the fundamental ones. The number 256 then appears as 2^(2^3), which the framework reads as 2^(2^D) where D is the dimension count 3. This structural match is part of a certificate that bundles the five-axiom count, the power set value, and the exponent form into one verified object.

In Recognition Science, this is not a claim about the physical universe. It is a claim about a formal definition inside the framework's own mathematics. The theorem does not say that any actual set in the world has 256 subsets, nor that the ZF axioms are the only possible foundation. It does not derive the number 256 from physical principles, and it does not imply that three-dimensional space must have 256 of anything. The declaration establishes a consistent, machine-checked arithmetic fact within the framework's chosen formal setting, nothing more and nothing less.

theorem powerSetQ3_eq_256 : powerSetQ3 = 256 := by decide

inductive FundamentalZFAxiom where
  | extensionality | pairing | union | powerSet | infinity
  deriving DecidableEq, Repr, BEq, Fintype

/-- 256 = 2^(2^D). -/
theorem powerSetQ3_2_2D : powerSetQ3 = 2 ^ (2 ^ 3) := by decide

What this page does not claim

The theorem does not claim that any physical set has 256 subsets. It does not derive the number 256 from physical principles or experimental measurement. It does not claim that the five chosen ZF axioms are the only possible foundation for mathematics.

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/SetTheoryFromRS.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.

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