SetTheoryCert
SetTheoryCert packages three equalities confirming that the recognition lattice Q₃ induces exactly five fundamental ZF axioms and a power set of size 256 matching 2^(2^3). Foundations researchers verifying the discrete lattice origin of classical set theory would cite this when checking the RS-to-ZF bridge. The declaration is a plain structure definition that records the three facts directly from the upstream inductive type and power-set definition.
claimA structure $C$ whose fields certify that the cardinality of the set of fundamental ZF axioms is 5, that the power set of the recognition lattice $Q_3$ has cardinality 256, and that this cardinality equals $2^{(2^3)}$.
background
The module derives set theory from the Recognition Science lattice Q₃. FundamentalZFAxiom is the inductive type whose five constructors are extensionality, pairing, union, powerSet and infinity. powerSetQ3 is the definition 2^8. The module states that these five axioms are the core ZF fragment used in RS and that the power set of Q₃ equals 256, matching the eight-tick structure 2^(2^3).
proof idea
The declaration is a structure definition with an empty proof body. It simply records the three field equalities supplied by the upstream definitions of FundamentalZFAxiom and powerSetQ3.
why it matters in Recognition Science
The structure supplies the type for the downstream definition setTheoryCert, which instantiates it with explicit equalities. It confirms that the recognition lattice produces the standard five ZF axioms and the expected power-set cardinality 2^8, consistent with the RS eight-tick octave and the identification of five axioms with configDim D.
scope and limits
- Does not derive the ZF axioms from the recognition composition law.
- Does not address the four non-fundamental ZF axioms.
- Does not construct a model of set theory.
- Does not compute the power-set cardinality from the J-cost function.
formal statement (Lean)
37structure SetTheoryCert where
38 five_axioms : Fintype.card FundamentalZFAxiom = 5
39 power_set_256 : powerSetQ3 = 256
40 structure_match : powerSetQ3 = 2 ^ (2 ^ 3)
41