Symbol Reference: ‹ ... › — Explicit Syntactic Quotation Brackets (Syntax Reflection)

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‹ ... › \(\lsaquo \dots \rsaquo\) — Explicit Syntactic Quotation Brackets (Syntax Reflection)

Reifies a syntactic formula or expression e into its canonical class-level AST / Gödel representation inside MKTD1 and MKTD2.

Field Mathematical & Kernel Specification (MKTD1 / MKTD2)
Symbol
‹ ... ›
\(\lsaquo \dots \rsaquo\)
Category Quotation & Antiquotation (Reflection Layer)
SML AST Constructor Kernel AST Symbol / Sort Constructor in typetheory.ML (Class_p / Truth_p / EXP / UCol)
Constructor Arguments Typed arguments as specified by the Signature / Sort field above
Metatype / Signature Expression → Class (or Fm → Class)
Binder Status See Category & Signature above
Defining Kernel Entries (ML Labels) See Related Symbols & Kernel Entries below.
LaTeX Form \lsaquo \dots \rsaquo
How to Understand It

In the Morse-Kelley + Tarski + Dunaev (MKTD1 / MKTD2) foundational framework, ‹𝛂› denotes explicit syntactic quotation (Corner / Guillemet Quotation). Rather than confusing a formula 𝛂 : Fm (or proposition P : Prop) with its syntactic code inside the universe of classes, quotation explicitly bridges the syntactic level and the object-class level:

  • No Implicit Confusion of Syntax and Semantics: A bare formula 𝛂 asserts truth when proved as a theorem, whereas ‹𝛂› denotes the structured syntactic object (or class code via Enc(𝛂)) representing the AST of 𝛂.
  • Compositional Quasi-Quotation: Inside a quoted expression ‹ ... ›, logical connectives and binders are quoted as AST constructors unless spliced via antiquotation ⟨ ... ⟩.
  • Role in Reflection & Incompleteness: Used throughout Section AxmSoundnessOfBaseTheory (3.72)–ThmDunaevSecondProjection (3.78), Section 4 (Metalogical Laws), and Section 5 (Löb's Theorem ThmLoeb (5.01) and Gödel's Second Incompleteness Theorem ThmSecondIncompleteness (5.02)) to state provability 𝒯 ⊢ ‹𝛂› cleanly.
Canonical Law / Usage (𝒯 ⊢ ‹𝛂›) ∧ (𝒯 ⊢ ‹𝛃›) ⇒ (𝒯 ⊢ ‹𝛂 ∧ 𝛃›)
Related Symbols & Kernel Entries