feat(ModelTheory): construct a counterexample to the two-cardinal phenomenon - #42190
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PR summary e3821a10c3Import changes exceeding 2%
|
| File | Base Count | Head Count | Change |
|---|---|---|---|
| Mathlib.ModelTheory.Semantics | 715 | 767 | +52 (+7.27%) |
| Mathlib.ModelTheory.ElementaryMaps | 799 | 814 | +15 (+1.88%) |
Import changes for all files
| Files | Import difference |
|---|---|
Mathlib.ModelTheory.Fraisse Mathlib.ModelTheory.Order |
9 |
Mathlib.ModelTheory.Algebra.Field.CharP |
12 |
14 filesMathlib.ModelTheory.Bundled Mathlib.ModelTheory.Complexity Mathlib.ModelTheory.DirectLimit Mathlib.ModelTheory.ElementaryMaps Mathlib.ModelTheory.ElementarySubstructures Mathlib.ModelTheory.Equivalence Mathlib.ModelTheory.FinitelyGenerated Mathlib.ModelTheory.Graph Mathlib.ModelTheory.PartialEquiv Mathlib.ModelTheory.Satisfiability Mathlib.ModelTheory.Skolem Mathlib.ModelTheory.Substructures Mathlib.ModelTheory.Topology.Types Mathlib.ModelTheory.Types |
15 |
Mathlib.ModelTheory.Algebra.Ring.FreeCommRing |
16 |
Mathlib.ModelTheory.Algebra.Field.Basic |
28 |
Mathlib.ModelTheory.Ultraproducts |
49 |
Mathlib.ModelTheory.Algebra.Ring.Basic Mathlib.ModelTheory.Arithmetic.Presburger.Basic |
50 |
3 filesMathlib.ModelTheory.Definability Mathlib.ModelTheory.Quotients Mathlib.ModelTheory.Semantics |
52 |
Mathlib.SetTheory.Cardinal.DirectLimit (new file) |
777 |
Mathlib.ModelTheory.ElementaryChain (new file) |
824 |
Mathlib.ModelTheory.DefinablyFull (new file) |
862 |
Declarations diff (regex)
+ DefinablyFull
+ ElementaryChain
+ Limit
+ UnaryDefinablyFull
+ card_le
+ coe_natLERec
+ definablyFull_of_unary
+ encard_realizations_eq_coe_iff
+ equivOfForallLE
+ exists_elementaryExtension_card_eq_with_full_unary_realizations
+ exists_finite_common_stage
+ exists_model_of_card_definably_full
+ exists_toLimit
+ finOneRealizationsEquiv
+ iExsAtLeast
+ iExsAtMost
+ iExsExactly
+ iSup_lift_mk_le_mk
+ infinite_realizations_iff
+ le_encard_iff_exists_injection_fin
+ lhomWithConstantsMap_isExpansionOn_of_eq
+ liftElementary
+ limitEquiv
+ limit_models
+ mk_eq_iSup_lift_mk
+ mk_eq_of_forall_lift_mk_eq
+ mk_le_of_aleph0_le
+ mk_le_sum
+ mk_limit_eq_of_forall_lift_mk_eq
+ mk_limit_le_of_lift_mk_le
+ mk_realizations_le
+ natLERec
+ natLERec_self
+ natLERec_succ
+ natLERec_trans
+ ofNatSucc
+ ofNatSucc_map_succ
+ realizationsEmbedding
+ realize_iExsAtLeast
+ realize_iExsAtLeast_iff_exists_injective
+ realize_iExsAtMost
+ realize_iExsAtMost_iff_not_exists_injection
+ realize_iExsAtMost_iff_not_iExsAtLeast_succ
+ realize_iExsExactly
+ realize_iExsExactly_one_iff_iExsUnique
+ realize_iExsExactly_zero
+ realize_toLimit
+ toEmbeddingDirectedSystem
+ toLimit
+ toLimitElementary
+ toLimit_map
++ card_le_withConstants
You can run this locally as follows
## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>
## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.
Declarations diff (Lean)
✅ Lean-aware diff — post-build, computed from the Lean environment (commit
e3821a1).
- +74 new declarations
- −0 removed declarations
+DirectLimit.equivOfForallLE
+DirectLimit.iSup_lift_mk_le_mk
+DirectLimit.mk_eq_iSup_lift_mk
+DirectLimit.mk_eq_of_forall_lift_mk_eq
+DirectLimit.mk_le_of_aleph0_le
+DirectLimit.mk_le_sum
+FirstOrder.Language.BoundedFormula.card_le_withConstants
+FirstOrder.Language.DefinablyFull
+FirstOrder.Language.ElementaryChain
+FirstOrder.Language.ElementaryChain.Limit
+FirstOrder.Language.ElementaryChain.carrier
+FirstOrder.Language.ElementaryChain.casesOn
+FirstOrder.Language.ElementaryChain.ctorIdx
+FirstOrder.Language.ElementaryChain.exists_finite_common_stage
+FirstOrder.Language.ElementaryChain.exists_toLimit
+FirstOrder.Language.ElementaryChain.liftElementary
+FirstOrder.Language.ElementaryChain.limit_models
+FirstOrder.Language.ElementaryChain.map
+FirstOrder.Language.ElementaryChain.map.congr_simp
+FirstOrder.Language.ElementaryChain.map_map
+FirstOrder.Language.ElementaryChain.map_self
+FirstOrder.Language.ElementaryChain.mk
+FirstOrder.Language.ElementaryChain.mk.congr_simp
+FirstOrder.Language.ElementaryChain.mk.inj
+FirstOrder.Language.ElementaryChain.mk.injEq
+FirstOrder.Language.ElementaryChain.mk.noConfusion
+FirstOrder.Language.ElementaryChain.mk.sizeOf_spec
+FirstOrder.Language.ElementaryChain.mk_limit_eq_of_forall_lift_mk_eq
+FirstOrder.Language.ElementaryChain.mk_limit_le_of_lift_mk_le
+FirstOrder.Language.ElementaryChain.noConfusion
+FirstOrder.Language.ElementaryChain.noConfusionType
+FirstOrder.Language.ElementaryChain.ofNatSucc
+FirstOrder.Language.ElementaryChain.ofNatSucc_map_succ
+FirstOrder.Language.ElementaryChain.realize_toLimit
+FirstOrder.Language.ElementaryChain.rec
+FirstOrder.Language.ElementaryChain.recOn
+FirstOrder.Language.ElementaryChain.struc
+FirstOrder.Language.ElementaryChain.toEmbeddingDirectedSystem
+FirstOrder.Language.ElementaryChain.toLimit
+FirstOrder.Language.ElementaryChain.toLimitElementary
+FirstOrder.Language.ElementaryChain.toLimit_map
+FirstOrder.Language.ElementaryEmbedding.encard_realizations_eq_coe_iff
+FirstOrder.Language.ElementaryEmbedding.infinite_realizations_iff
+FirstOrder.Language.ElementaryEmbedding.mk_realizations_le
+FirstOrder.Language.ElementaryEmbedding.natLERec
+FirstOrder.Language.ElementaryEmbedding.natLERec.congr_simp
+FirstOrder.Language.ElementaryEmbedding.natLERec_self
+FirstOrder.Language.ElementaryEmbedding.natLERec_succ
+FirstOrder.Language.ElementaryEmbedding.natLERec_trans
+FirstOrder.Language.ElementaryEmbedding.realizationsEmbedding
+FirstOrder.Language.Formula.card_le
+FirstOrder.Language.Formula.card_le_withConstants
+FirstOrder.Language.Formula.finOneRealizationsEquiv
+FirstOrder.Language.Formula.iExsAtLeast
+FirstOrder.Language.Formula.iExsAtLeast.congr_simp
+FirstOrder.Language.Formula.iExsAtMost
+FirstOrder.Language.Formula.iExsAtMost.congr_simp
+FirstOrder.Language.Formula.iExsExactly
+FirstOrder.Language.Formula.iExsExactly.congr_simp
+FirstOrder.Language.Formula.realize_iExsAtLeast
+FirstOrder.Language.Formula.realize_iExsAtLeast_iff_exists_injective
+FirstOrder.Language.Formula.realize_iExsAtMost
+FirstOrder.Language.Formula.realize_iExsAtMost_iff_not_exists_injection
+FirstOrder.Language.Formula.realize_iExsAtMost_iff_not_iExsAtLeast_succ
+FirstOrder.Language.Formula.realize_iExsExactly
+FirstOrder.Language.Formula.realize_iExsExactly_one_iff_iExsUnique
+FirstOrder.Language.Formula.realize_iExsExactly_zero
+FirstOrder.Language.LHom.lhomWithConstantsMap_isExpansionOn_of_eq
+FirstOrder.Language.Theory.ModelType.of.congr_simp
+FirstOrder.Language.Theory.exists_elementaryExtension_card_eq_with_full_unary_realizations
+FirstOrder.Language.Theory.exists_model_of_card_definably_full
+FirstOrder.Language.UnaryDefinablyFull
+FirstOrder.Language.definablyFull_of_unary
+Set.le_encard_iff_exists_injection_finIncrease in strong tech debt: (relative, absolute) = (1.00, 0.00)
| Current number | Change | Type (strong) |
|---|---|---|
| 500 | 1 | erw |
Increase in weak tech debt: (relative, absolute) = (3.00, 0.00)
| Current number | Change | Type (weak) |
|---|---|---|
| 5026 | 3 | exposed public sections |
Current commit e3821a10c3
Reference commit 5b1613e1da
This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary
- The
relativevalue is the weighted sum of the differences with weight given by the inverse of the current value of the statistic. - The
absolutevalue is therelativevalue divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).
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Show the existence of a model of a complete theory with infinite models in a countable language where every infinite definable set has the same cardinality of the model itself for every infinite cardinality, which is a counterexample to the two-cardinal phenomenon.
This result is an ingredient in the proof of Morley’s categoricity theorem. Let$T$ be a complete theory in a countable language with an infinite model, and let $\kappa$ be uncountable. If $T$ is $\omega$ -stable and has a Vaughtian pair, the pair can be transferred to a $(\kappa, \aleph_0)$ -model of $T$ : a model of cardinality $\kappa$ containing a countably infinite definable set. Such a model cannot be isomorphic to the model constructed in this PR, since cardinalities of definable sets are preserved by isomorphisms. Thus, an $\omega$ -stable $\kappa$ -categorical theory has no Vaughtian pairs. Together with the converse and the fact that neither implication depends on the particular uncountable cardinal $κ$ , this yields the categoricity transfer in Morley’s theorem.
The present proof still exposes some foundational infrastructure and auxiliary lemmas that should likely be extracted and developed separately. This PR will remain a draft until this supporting work has been fully implemented.
AI disclosure: Most declarations, comments, and documentation in this PR were generated with Codex. Some proofs were structurally simplified using the
/golfcommand from Lean 4 Skills. I reviewed the resulting code and documentation and can vouch for all submitted content.