Infinitary Logic in Lean 4

A Lean 4 formalization of infinitary logic (L∞ω and Lω₁ω), Scott sentences, and classical results in infinitary model theory, building on Mathlib.

• Project page
• Blueprint (web) · Blueprint (pdf)
• API docs · Dependency graph
• Releases · How to cite

Results

The blueprint states these precisely and narrates their proofs; this table names the endpoints and the hypotheses that matter.

Scott analysis and Karp's theorem

Result	In Lean	Scope
Every countable structure has a Scott sentence characterizing it up to isomorphism among countable structures	scottSentence	countable relational language
Scott rank is a countable ordinal	scottRank
Back-and-forth equivalence at all ordinals characterizes L∞ω-equivalence	karp_theorem_w

Model theory of Lω₁ω

Result	In Lean	Scope
Model existence for consistency properties, with omitting types and Karp completeness	model_existence	countable language
ℶ_ω₁ is a Hanf bound for every Lω₁ω-sentence	morley_hanf	arbitrary language, no side hypotheses
The bound is sharp: Hanf(Lω₁ω) = ℶ_ω₁	Lomega1omegaHanfNumber_eq_beth_omega1
Models of every infinite size realizing only countably many complete types	exists_small_model_of_hasArbLargeModels	arbitrary languages
Small models lie in complete subclasses; κ-categorical sentences have κ-categorical complete completions	exists_complete_sentence_of_lomega1omegaSmall, exists_complete_kCategorical_of_hasArbLargeModels	countable relational vocabulary

Interpolation

All three are sentence-level. The theory-level analogues are false for Lω₁ω and are never claimed.

Result	In Lean	Scope
Craig — interpolants whose function and relation symbols each lie in the intersection of the two roots' occurrence sets; also the PC-separation form	craig_interpolation, craig_pcSeparation	arbitrary language, no hypotheses
Lyndon — the polarity refinement: positively occurring relation symbols bounded by the roots' shared positive occurrences, negatively occurring ones by their shared negative occurrences	lyndon_interpolation	arbitrary language; relation polarity, logical equality — equality belongs to neither polarity class and is unconstrained, so López-Escobar's clause (.3) is not claimed
Malitz — the quantifier-class refinement: an entailment with universal consequent has a universal interpolant	malitz_interpolation	universal consequent, relational language of arbitrary cardinality

Definability and undefinability

Result	In Lean	Scope
Chains of every countable length force a model carrying a relation-preserving map from ℚ (Marker 4.26)	exists_model_relPreserving	arbitrary language; raw positive form, no injectivity claimed
A uniform countable bound on the order types of well-ordered models (Marker 4.27)	wellOrder_type_boundedness
No Lω₁ω-sentence has as models exactly the well-orders	wellOrdering_undefinable
López–Escobar — a class of coded countable structures is Borel and isomorphism-invariant iff it is the model class of a single Lω₁ω-sentence; equivalently the invariant Borel classes are exactly the range of ModelsOf	lopezEscobar_iff, lopezEscobar_action_iff	countable relational vocabulary
The coded well-order class WO is not Borel in the logic space	wellOrderClass_not_measurableSet	the cheap half of Π¹₁-completeness; many-one hardness not claimed

Descriptive set theory

Result	In Lean	Scope
Silver's theorem — a Borel equivalence relation on a Polish space has countably many classes or a perfect set of pairwise-inequivalent points	gandy_harrington_for_relation	via Miller's category route (G₀-dichotomy, Kuratowski–Ulam, Mycielski — all formalized here)
The Silver–Burgess dichotomy — on a standard Borel space the quotient is ≤ ℵ₀ or exactly 2^ℵ₀	silverBurgessDichotomy
Morley counting — countable models of an Lω₁ω-sentence number ≤ ℵ₁ or exactly 2^ℵ₀	morley_counting	parametrized by the dichotomy, which this repository proves
Isomorphism is Borel under bounded Scott height	iso_borel_of_bounded_scottHeight

Fragments and admissibility

An honest coded-fragment interface, with the HF fragment as its regression instance.

Result	In Lean	Scope
Coded-family presentations — a family is a code together with its decoding, its index type supplied by the code and carrying an explicit Encodable, and its infinitary status a certificate the presentation grants	AdmissiblePresentation, CodedFamily
Honest coded closure — a Fragment closed upward under exactly the families a presentation certifies, and under nothing else. Deliberately carries no compactness data: compactness is a theorem with hypotheses, not a field	AdmissibleFragment
The HF fragment — the first-order image inside Lω₁ω, L_HF = L_ωω, as an instance with no adapter and no widening; its coded families are uninhabited, so both upward obligations are vacuous	hfFragment, hfAdmissibleFragment	sentence slice proved equal to finitaryFragment
HF compactness, derived from Mathlib's first-order compactness rather than assumed	finitaryFragment_compact	the semantic step is at Language.{0,0}; the syntax layer is universe-polymorphic

Barwise compactness and the Nadel bound are not proved, and are not claimed. The interfaces carrying them (Admissible/Barwise/*, Admissible/Compactness.lean, Admissible/Nadel.lean, and the placeholders AdmissibleFragmentCore.hf, FullBarwiseFragment, FiniteCompactFragment.CodedIn) package a hypothesis rather than discharging it, are labelled as such in source, and are being replaced by the interface above. Progress is tracked in #18–#20. Malitz's relative preservation theorem (4.6) is likewise not proved (#41).

Getting Started

git clone https://github.com/cameronfreer/infinitary-logic.git && cd infinitary-logic
lake build

To use in your own project, add the dependency to your lakefile and import a bundle:

import InfinitaryLogic.Core         -- syntax, semantics, Scott, Karp
import InfinitaryLogic.Countable    -- model existence, LS, Hanf, EM chain
import InfinitaryLogic.Admissible   -- coded-fragment interface, HF; legacy conditional interfaces
import InfinitaryLogic.Descriptive  -- descriptive set theory of model classes
import InfinitaryLogic.All          -- all of the above
import InfinitaryLogic.Conditional  -- Silver chain + Morley-Hanf theorem (both proved)
import InfinitaryLogic.Everything   -- everything including Conditional and legacy off-path modules

import InfinitaryLogic loads the default surface (InfinitaryLogic.All). Work-in-progress frontier modules live in the separate non-default InfinitaryLogicWIP target, so they never enter it.

Repository Guide

Directory	Contents
Lomega1omega/	the Lω₁ω layer over Mathlib's fixed-carrier infinitary syntax — semantics, operations, fragments, polarity, quantifier rank
Scott/, Karp/	atomic diagrams, back-and-forth equivalence, Scott formulas and sentences, rank and height; Karp's theorem
Methods/	the proof engines: the Henkin/consistency-property kernel, interpolation, the well-ordering machine, López–Escobar, Ehrenfeucht–Mostowski
ModelTheory/	Löwenheim–Skolem, Hanf numbers and the Hanf spectrum, small models, counting
Admissible/	the coded-fragment interface and HF (above), plus the legacy conditional scaffolding
Descriptive/	the Borel structure space and a reusable descriptive-set-theory library — Cantor-antichain extraction, Cantor-subcopy stabilization of Borel data, Mycielski, Kuratowski–Ulam, the G₀ dichotomy and fusion
Combinatorics/	infinite Ramsey and the bounded finite-arity Erdős–Rado chain
Conditional/	the Silver and Morley–Hanf chains, including the unconditional morley_hanf endpoint (the directory name is historical)

Verification

The tree is sorry-free, and the headline results depend on exactly propext, Classical.choice and Quot.sound. CI builds the public and frontier targets and enforces both the proof boundary and the axiom boundary on every commit.

Three dependency-cone guards additionally certify proof architecture, where an axiom scan cannot reach: that the Henkin route consumes no maximal-consistency machinery, that the Morley–Hanf cone avoids the legacy Erdős–Rado ladder, and that HF compactness genuinely consumes Mathlib's compactness theorem while touching none of the legacy admissible structures.

Operational notes on building and releasing are in docs/build-and-release-notes.md.

References

• Barwise, J. (1975). Admissible Sets and Structures. Springer-Verlag.
• Karp, C. R. (1964). Languages with Expressions of Infinite Length. North-Holland.
• Karp, C. R. (1965). Finite-Quantifier Equivalence. In The Theory of Models, 407–412.
• Keisler, H. J. (1971). Model Theory for Infinitary Logic. North-Holland.
• López-Escobar, E. G. K. (1965). An interpolation theorem for denumerably long formulas. Fundamenta Mathematicae, 57, 253–272.
• Keisler, H. J. & Knight, J. F. (2004). Barwise: Infinitary Logic and Admissible Sets. Bulletin of Symbolic Logic, 10(1), 4–36.
• Marker, D. (2016). Lectures on Infinitary Model Theory. Cambridge University Press.
• Nadel, M. E. (1974). Scott sentences and admissible sets. Annals of Mathematical Logic, 7(2–3), 267–294.

License & Citation

Apache 2.0 licensed. See LICENSE for details.

@software{freer2026infinitary,
  author = {Cameron Freer},
  title = {Infinitary Logic in {Lean} 4},
  url = {https://github.com/cameronfreer/infinitary-logic},
  year = {2026}
}