ω / Nat \(\omega\) — Smallest Successor-Inductive Ordinal Constant (ω, AST Constructor: CpOmega)
0-ary class constant ω defined as the intersection of all successor-inductive classes Q (Entry DefInductiveClass (3.55)), making ω both the smallest successor-inductive ordinal and the ⊆-minimal successor-inductive class.
| Field | Mathematical & Kernel Specification (MKTD1 / MKTD2) |
|---|---|
| Symbol |
ω / Nat
\(\omega\)
|
| Category | Functional Symbol (0-ary Class Constant) |
| SML AST Constructor | HC CpOmega [in datatype Class_p, typetheory.ML:558] |
| Constructor Arguments | 0-ary constant (no arguments) |
| Metatype / Signature | Class |
| Binder Status | No — 0-ary Functional Symbol (Smallest Successor-Inductive Ordinal ω) |
| Defining Kernel Entries (ML Labels) | See Related Symbols & Kernel Entries below. |
| LaTeX Form | \omega |
| How to Understand It | SML AST Constructor: |
| Canonical Law / Usage | ∀(x:Class)((x ∈ ω) ⟷ ∀(Q:Class)(Ind(Q) → (x ∈ Q))) |
| Related Symbols & Kernel Entries |