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Trakhtenbrot's theorem: finite satisfiability is undecidable

Lax624099.Trakhtenbrot · concepts/Lax624099/Trakhtenbrot.lean · lax-624099

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    Natural Language Statement

    Theorem

    Finite satisfiability of first-order sentences is not decidable: no computable predicate on concrete encoded sentences agrees with FINSAT. This is the theorem of Trakhtenbrot (1950).

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

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    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.Classes
    5import Lax624099.Problems
    6import Lax624099.ValueInvention
    7import Lax624099.ClassRE
    8import Lax624099.FiniteSatisfiability
    9import Lax624099.Halting
    10import Lax624099.CodeHalting
    11import Lax624099.PostCorrespondence
    12import Lax624099.ConcreteInstances
    13import Lax904597.Machines
    14
    15/-!
    16---
    17title: Trakhtenbrot's theorem: finite satisfiability is undecidable
    18type: theorem
    19---
    20Finite satisfiability of first-order sentences is not decidable: no
    21computable predicate on concrete encoded sentences agrees with FINSAT. This
    22is the theorem of Trakhtenbrot (1950).
    23-/
    24
    25namespace Lax624099.Trakhtenbrot
    26
    27open FirstOrder FirstOrder.Language
    28open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.Classes
    29open Lax904597.Machines Lax624099.Problems Lax624099.ValueInvention Lax624099.ClassRE
    30open Lax624099.FiniteSatisfiability
    31open Lax624099.Halting Lax624099.CodeHalting Lax624099.PostCorrespondence
    32open Lax624099.ConcreteInstances
    33
    34/-- Trakhtenbrot's theorem. -/
    35axiom finsat_not_computable : ¬ComputablePred (DecisionProblem.toPred FINSAT finsatVocab)
    36
    37end Lax624099.Trakhtenbrot
    38
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