Mathematics Set Theory From Rs Power Set Q3 2 2 D

The power set of Q₃

In mathematics, the power set of a set is the collection of all its subsets. A set with three elements has eight subsets, and the set of those subsets has 256 members. This is simple combinatorics: 2 raised to the power 3 equals 8, and 2 raised to the power 8 equals 256.

Zermelo-Fraenkel set theory, the standard foundation of mathematics, includes an axiom guaranteeing that the power set of any set exists. The Recognition Science framework works with a specific three-element structure it calls the recognition lattice Q₃, and its machine-checked library of formal theorems contains a declaration, powerSetQ3_2_2D, that verifies the size of Q₃'s power set. The theorem states that this power set has 2^(2^3) members, which is 256.

In Recognition Science, the framework models this three-element lattice as the foundation of its recognition ledger, a discrete record of events. The theorem is a calculation: it confirms that the power set of the lattice has the same cardinality as the power set of any three-element set. The library checks this by direct computation, with no unproved assumptions and no axioms beyond the standard logical ones.

The declaration does not claim that Q₃ is the only structure with a 256-member power set, nor that the number 256 has any special significance beyond being the size of this particular power set. It does not assert that the Zermelo-Fraenkel axioms are derived from Recognition Science, only that five of them are used as a starting point. The theorem is a precise, narrow fact about a specific structure, not a claim about the nature of set theory as a whole.

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

What this page does not claim

The declaration does not claim that Q₃ is the only three-element set or that 256 is a special number beyond this context. It does not claim that the Zermelo-Fraenkel axioms are derived from Recognition Science, only that five are used. It does not claim any physical or empirical consequence from the size of this power set.

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)

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