StructuredSet
The declaration defines the structured set as the collection of positive reals where the defect functional vanishes. Researchers formalizing the Law of Existence cite this set when equating existence, zero defect, and the value 1. The definition is a direct set comprehension that applies the defect predicate to positive reals.
claimDefine the structured set by $S := {x ∈ ℝ | 0 < x ∧ defect(x) = 0}$.
background
The Law of Existence module states that a positive real x exists precisely when its defect vanishes. The defect functional is supplied by the upstream definition defect(x) := J(x), where J is the Recognition cost function. The module records the equivalences defect(x) = 0 ↔ x = 1 and existence ↔ defect collapse, with 1 as the unique point of vanishing defect.
proof idea
This is a direct definition by set comprehension. The body collects exactly those x that satisfy the two conjuncts 0 < x and defect x = 0; no lemmas or tactics are invoked.
why it matters in Recognition Science
The definition supplies the set object referenced by the complete law of existence theorem, which equates membership in the structured set with existence and with equality to 1. It also supports the economic inevitability theorem that identifies 1 as the unique defect minimizer. In the Recognition framework the set encodes the T5 J-uniqueness step at the level of sets, linking the forcing chain to the unique existent.
scope and limits
- Does not prove that the set equals the singleton {1}.
- Does not supply a membership decision procedure beyond the defect predicate.
- Does not extend the definition to non-positive reals.
formal statement (Lean)
155def StructuredSet : Set ℝ := {x : ℝ | 0 < x ∧ defect x = 0}
proof body
Definition body.
156