03-XX Mathematical logic and foundations
03-00 General reference works (handbooks, dictionaries, bibliographies, etc.)
03-01 Instructional exposition (textbooks, tutorial papers, etc.)
03-02 Research exposition (monographs, survey articles)
03-03 Historical {!must also be assigned at least one classification number from section 01}
03-04 Explicit machine computation and programs (not the theory of computation or programming)
03-06 Proceedings, conferences, collections, etc.
03A05 Philosophical and critical {For philosophy of mathematics, see also 00A30}
03Bxx General logic
03B05 Classical propositional logic
03B10 Classical first-order logic
03B15 Higher-order logic and type theory
03B20 Subsystems of classical logic (including intuitionistic logic)
03B22 Abstract deductive systems
03B25 Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10]
03B30 Foundations of classical theories (including reverse mathematics) [See also 03F35]
Foundation and axiomatics of classical theories
03B35 Mechanization of proofs and logical operations [See also 68T15]
03B40 Combinatory logic and lambda-calculus [See also 68N18]
/:> [See also 68N18]
03B42 Logic of knowledge and belief
03B44 Temporal logic
03B45 Modal logic {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45}
Modal and tense logic {For provability logic, see also 03F40}
03B46
Relevance and entailment
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects, see 03F52}
03B48 Probability and inductive logic [See also 60A05]
03B50 Many-valued logic
03B52 Fuzzy logic; logic of vagueness [See also 68T27, 68T37, 94D05]
/:> ; Logic of vagueness /:> 68T27, 68T37,
03B53 Logics admitting inconsistency (paraconsistent logics, discussive logics, etc.)
Paraconsistent logic
03B55 Intermediate logics
03B60 Other nonclassical logic
03B65 Logic of natural languages [See also 68T50, 91F20]
// 68T50, 91F20 ~ 68S05, 92K20
03B70 Logic in computer science [See also 68-XX]
Logic of programming [See also 68Q55, 68Q60]
03B80 Other applications of logic
03B99 None of the above, but in this section
03Cxx Model theory
03C05 Equational classes, universal algebra [See also 08Axx, 18C05]
/:> , 18C05
03C07 Basic properties of first-order languages and structures
03C10 Quantifier elimination, model completeness and related topics
/:> , Model completeness
03C13 Finite structures [See also 68Q15, 68Q19]
/:> [See also 68Q15, 68Q19]
03C15 Denumerable structures
03C20 Ultraproducts and related constructions
03C25 Model-theoretic forcing
03C30 Other model constructions
03C35 Categoricity and completeness of theories
03C40 Interpolation, preservation, definability
03C45 Classification theory, stability and related concepts
/:> Classification theory
03C50 Models with special properties (saturated, rigid, etc.)
03C52 Properties of classes of models
03C55 Set-theoretic model theory
03C57 Effective and recursion-theoretic model theory [See also 03D45]
/:> Effective and
03C60 Model-theoretic algebra [See also 08C10, 12Lxx, 13L05]
03C62 Models of arithmetic and set theory [See also 03Hxx]
03C64 Model theory of ordered structures; o-minimality
03C65 Models of other mathematical theories
03C68 Other classical first-order model theory
03C70 Logic on admissible sets
03C75 Other infinitary logic
03C80 Logic with extra quantifiers and operators [See also 03B42, 03B44, 03B45, 03B48]
/:> 03B42, 03B44, 03B48,
03C85 Second- and higher-order model theory
03C90 Nonclassical models (Boolean-valued, sheaf, etc.)
03C95 Abstract model theory
03C98 Applications of model theory [See also 03C60]
03C99 None of the above, but in this section
03Dxx Computability and recursion theory
/:> Computability and
03D03 Thue and Post systems, etc.
03D05 Automata and formal grammars in connection with logical questions [See also 68Q45, 68Q70, 68R15]
// 68Q45, 68Q70, 68R15 ~ 68Qxx
03D10 Turing machines and related notions [See also 68Q05]
03D15 Complexity of computation [See also 68Q15, 68Q17]
/:> 68Q17
03D20 Recursive functions and relations, subrecursive hierarchies
03D25 Recursively (computably) enumerable sets and degrees
/:> (Computably)
03D28 Other Turing degree structures
03D30 Other degrees and reducibilities
03D35 Undecidability and degrees of sets of sentences
03D40 Word problems, etc. [See also 06B25, 08A50, 20F10]
03D45 Theory of numerations, effectively presented structures [See also 03C57] {For intuitionistic and similar approaches, see 03F55}
/:> {For intuitionistic and similar approaches, see 03F55]
03D50 Recursive equivalence types of sets and structures, isols
03D55 Hierarchies
03D60 Computability and recursion theory on ordinals, admissible sets, etc.
/:> Computability and
03D65 Higher-type and set recursion theory
03D70 Inductive definability
03D75 Abstract and axiomatic computability and recursion theory
/:> Computability and
03D80 Applications of computability and recursion theory
03D99 None of the above, but in this section
03Exx Set theory
/:< [See also 04-XX]
03E02 Partition relations
03E04 Ordered sets and their cofinalities; pcf theory
03E05 Other combinatorial set theory
/:< [See also 04A20] /:> other
03E10 Ordinal and cardinal numbers
/:< [See also 04A10]
03E15 Descriptive set theory [See also 28A05, 54H05]
/:< 04A10
03E17 Cardinal characteristics of the continuum
03E20 Other classical set theory (including functions, relations, and set algebra)
/:> (Including functions, relations, and set algebra)
03E25 Axiom of choice and related propositions
/:< [See also 04A25]
03E30 Axiomatics of classical set theory and its fragments
03E35 Consistency and independence results
03E40 Other aspects of forcing and Boolean-valued models
03E45 Inner models, including constructibility, ordinal definability, and core models
/:> Inner models, including // core models ~ related notions
03E47 Other notions of set-theoretic definability
03E50 Continuum hypothesis and Martin's axiom
/:< [See also 04A30, 54A25]
03E55 Large cardinals
03E60 Determinacy principles
Determinacy and related principles which contradict the axiom of choice
03E65 Other hypotheses and axioms
03E70 Nonclassical and second-order set theories
03E72 Fuzzy set theory
Fuzzy sets [See mainly 04A72]
03E75 Applications of set theory
/:> Of set theory
03E99 None of the above, but in this section
03Fxx Proof theory and constructive mathematics
03F03 Proof theory, general
03F05 Cut-elimination and normal-form theorems
03F07 Structure of proofs
03F10 Functionals in proof theory
03F15 Recursive ordinals and ordinal notations
03F20 Complexity of proofs
03F25 Relative consistency and interpretations
03F30 First-order arithmetic and fragments
03F35 Second- and higher-order arithmetic and fragments [See also 03B30]
// 03B30 ~ 03E30, 03E70
03F40 G\"odel numberings in proof theory
03F45 Provability logics and related algebras (e.g., diagonalizable algebras) [See also 03B45, 03G25, 06E25]
03F50 Metamathematics of constructive systems
03F52 Linear logic and other substructural logics [See also 03B47]
03F55 Intuitionistic mathematics
03F60 Constructive and recursive analysis [See also 03B30, 03D45, 26E40, 46S30, 47S30]
/:> 03B30, 03D45,
03F65 Other constructive mathematics [See also 03D45]
// 03D45 ~ 26E40, 46S30, 47S30
03F99 None of the above, but in this section
03Gxx Algebraic logic
03G05 Boolean algebras [See also 06Exx]
03G10 Lattices and related structures [See also 06Bxx]
03G12 Quantum logic [See also 06C15, 81P10]
/:> 06C15,
03G15 Cylindric and polyadic algebras; relation algebras
03G20 Lukasiewicz and Post algebras [See also 06D25, 06D30]
03G25 Other algebras related to logic [See also 03F45, 06D20, 06E25, 06F35]
/:> 03F45, 06D20, 06E25,
03G30 Categorical logic, topoi [See also 18B25, 18C05, 18C10]
/:> 18C05, 18C10,
03G99 None of the above, but in this section
03Hxx Nonstandard models [See also 03C62]
03H05 Nonstandard models in mathematics [See also 26E35, 28E05, 30G06, 46S20, 47S20, 54J05]
03H10 Other applications of nonstandard models (economics, physics, etc.)
03H15 Nonstandard models of arithmetic [See also 11U10, 12L15, 13L05]
03H99 None of the above, but in this section