FunctionaL-City

Civic Set Ordinance Book

Subdivision: CivicAlgebraicInfrastructure.Foundations.CivicSet
Companion files:

CivicSet.fs — ordinance code
CivicSet.fsx — interactive case law

Preamble

This ordinance establishes the foundations of Civic Sets within the algebraic infrastructure. A CivicSet is a narratable domain—defined by its symbolic signage, logical overlays, and metadata zoning. Each set is a lawful citizen of the infrastructure, equipped with sample elements, membership predicates, and civic annotations for remixers.

Purpose

Define lawful domains with symbolic and logical signage.
Provide sample elements for inspection and onboarding.
Declare ordering signage (min, max, compare).
Integrate FOL overlays for logical reasoning.
Maintain a metadata registry for set-theoretic and logical properties.
Enable closure, implication, and equivalence overlays for civic reasoning.

Ordinance Sections

1. Domain Model

Symbol: Unicode or civic-native signage (e.g. ℕ, ℤ, ℝ, ℂ).
Formula: Optional FOL signage overlay (Formula<'Symbol>).
Contains: Membership predicate ('Concrete -> bool).
Elements: Sample enumeration (seq<'Concrete>), used for onboarding and inspection.

2. Ordering Signage

Compare: Optional civic-native comparison function.
Min / Max: Declared extrema, if applicable.
Lawfulness: Ordering signage must respect domain semantics (e.g. ℂ is unordered).

3. Metadata

SetTheoreticMetadata:
- Cardinality: Finite, ℵ₀, Continuum, or Other.
- Countability: Countable or Uncountable.
- OrderType: TotalOrder, PartialOrder, or Unordered.
FOLMetadata:
- Decidability: Decidable, SemiDecidable, Undecidable.
- Enumerability: Enumerable or NonEnumerable.
Other CivicSetMetadataItem:
- Provenance: e.g. “Defined in ZFC”, “Derived from Peano axioms”.
- Tag: Free-form signage.
- Note: Civic commentary.
- Custom: Extension point (string * string).

| Metadata Item         | Source Ordinance         | Derivation Path                                      | Remix Rights             |
|-----------------------|--------------------------|------------------------------------------------------|--------------------------|
| SetTheoretic          | ZFC, Cantor              | Declared via constructor or derived from set traits  | Declare, Derive          |
| FOL                   | Peano, FOL               | Derived from formula structure and quantifier scope  | Derive                   |
| Provenance            | Civic signage            | Free-form declaration of origin                      | Declare                  |
| Tag                   | Civic signage            | Free-form civic-native annotation                    | Declare, Extend          |
| Note                  | Civic signage            | Commentary block or onboarding signage               | Declare                  |
| Custom (key, value)   | Extension ordinance      | Declared by remixers for domain-specific overlays    | Declare, Extend          |

type CivicSetMetadataItem =
    | SetTheoretic of SetTheoreticMetadata
    | FOL of FOLMetadata
    | Provenance of string
    | Tag of string
    | Note of string
    | Custom of string * string

Zoning Map

┌──────────────────────────────┐
│ ICivicSet<'Concrete,'Symbol> │
└────────────┬─────────────────┘
             │
             ▼
┌──────────────────────────────┐
│ Symbol : string option       │
│ Formula : Formula<'Symbol>   │
│ Contains : 'Concrete -> bool │
│ Elements : seq<'Concrete>    │
└────────────┬─────────────────┘
             │
             ▼
┌──────────────────────────────────────────────────┐
│ Compare : option<'Concrete -> 'Concrete -> int>  │
│ Min / Max : option<'Concrete>                    │
└────────────┬─────────────────────────────────────┘
             │
             ▼
┌──────────────────────────────────────┐
│ Metadata : CivicSetMetadataItem list │
└────────────┬─────────────────────────┘
             │
             ▼
┌──────────────────┐
│ Logical Overlays │
│ IsClosedUnder    │
│ Implies          │
│ EquivalentTo     │
└──────────────────┘

🏷️ Glossary

CivicSetMetadataItem

Definition: A union of signage overlays annotating a civic set with logical, set-theoretic, and provenance metadata.
Variants: SetTheoretic, FOL, Provenance, Tag, Note, Custom
Signage Overlay: CivicSetMetadataItem scaffolds remix inspection, audit overlays, and symbolic lineage guarantees.

SetProvenance

Definition: A symbolic record of how a civic set was derived, constructed, or imported.
Variants: FromUnion, FromConstructor, FromAxiom, FromSymbolic, etc.
Signage Overlay: SetProvenance is the civic ancestry trail. It scaffolds timestamp semantics and lineage inspection.

ICivicSet<’Concrete,’Symbolic’>

Definition: A civic interface for symbolic sets with metadata, provenance, and logical overlays.
Signage Overlay: ICivicSet is the zoning charter for symbolic set infrastructure. It scaffolds lawful containment, symbolic reasoning, and audit overlays.

unionLiftedSets

Definition: A constructor that lifts two civic sets into a symbolic union with derived provenance and optional formula.
Signage Overlay: unionLiftedSets scaffolds remix-safe set composition, symbolic lineage, and onboarding clarity.

Case Law (see `.fsx`)

Example: Defining ℕ with Symbol = ℕ, Elements = Seq.initInfinite id, and Cardinality = ℵ₀.
Example: Declaring ℂ as Unordered, Uncountable, with no Compare.
Example: Annotating ℚ with FOLMetadata = { Decidable; Enumerable }.
Example: Testing closure: ℤ.IsClosedUnder (fun s -> s) returns true.
Example: Inspecting signage: printfn “ℝ formula: %A” reals.Formula.

Commentary

This ordinance is designed to be remixable:

Citizens may define new domains with lawful signage.
Metadata is optional but encouraged for civic clarity.
Logical overlays are declarative, not enforced—remixers may opt in.
Civic sets may serve as onboarding signage, audit overlays, or dispatch zones.
FOL integration enables symbolic reasoning across domains.

Crosslinks

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