Preliminaries to classical first-order model theory.

*(English)*Zbl 1276.03030Summary: First of a series of articles laying down the bases for classical first-order model theory. These articles introduce a framework for treating arbitrary languages with equality. This framework is kept as generic and modular as possible: both the language and the derivation rule are introduced as a type, rather than a fixed functor; definitions and results regarding syntax, semantics, interpretations and sequent derivation rules, respectively, are confined to separate articles, to mark out the hierarchy of dependences among different definitions and constructions.

As an application limited to countable languages, satisfiability theorem and a full version of the Gödel completeness theorem are delivered, with respect to a fixed, remarkably thrifty, set of correct rules. Besides the self-referential significance for the Mizar project itself of those theorems being formalized with respect to a generic, equality-furnished, countable language, this is the first step to work out other milestones of model theory, such as Löwenheim-Skolem and compactness theorems. Being the receptacle of all results of broader scope stemmed during the various formalizations, this first article stays at a very generic level, with results and registrations about objects already in the Mizar Mathematical Library.

Without introducing the Language structure yet, three fundamental definitions of wide applicability are also given: the ‘unambiguous’ attribute, the functor ‘-multiCat’, which is the iteration of ‘\^{}’ over a FinSequence of FinSequence, and the functor SubstWith, which realizes the substitution of a single symbol inside a generic FinSequence.

As an application limited to countable languages, satisfiability theorem and a full version of the Gödel completeness theorem are delivered, with respect to a fixed, remarkably thrifty, set of correct rules. Besides the self-referential significance for the Mizar project itself of those theorems being formalized with respect to a generic, equality-furnished, countable language, this is the first step to work out other milestones of model theory, such as Löwenheim-Skolem and compactness theorems. Being the receptacle of all results of broader scope stemmed during the various formalizations, this first article stays at a very generic level, with results and registrations about objects already in the Mizar Mathematical Library.

Without introducing the Language structure yet, three fundamental definitions of wide applicability are also given: the ‘unambiguous’ attribute, the functor ‘-multiCat’, which is the iteration of ‘\^{}’ over a FinSequence of FinSequence, and the functor SubstWith, which realizes the substitution of a single symbol inside a generic FinSequence.

##### MSC:

03C07 | Basic properties of first-order languages and structures |

03B35 | Mechanization of proofs and logical operations |

68T15 | Theorem proving (deduction, resolution, etc.) (MSC2010) |

##### Keywords:

Mizar Mathematical Library##### Software:

Mizar
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\textit{M. Caminati}, Formaliz. Math. 19, No. 3, 155--167 (2011; Zbl 1276.03030)

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##### References:

[1] | Broderick Arneson and Piotr Rudnicki. Recognizing chordal raphs: Lex BFS and MCS. Formalized Mathematics, 14(4):187-205, 2006, doi:10.2478/v10037-006-0022-z. |

[2] | Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990. |

[3] | Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. · Zbl 1364.68157 |

[4] | Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589-593, 1990. |

[5] | Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990. |

[6] | Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213-225, 1992. |

[7] | Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990. |

[8] | Grzegorz Bancerek and Yasunari Shidama. Introduction to matroids. Formalized Mathematics, 16(4):325-332, 2008, doi:10.2478/v10037-008-0040-0. |

[9] | Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996. |

[10] | Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245-254, 1990. |

[11] | Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990. |

[12] | Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990. |

[13] | Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990. |

[14] | Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990. |

[15] | Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990. |

[16] | Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990. |

[17] | Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990. |

[18] | Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990. |

[19] | Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990. |

[20] | M. Lothaire. Algebraic combinatorics on words. Cambridge Univ Pr, 2002. · Zbl 1001.68093 |

[21] | Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990. |

[22] | Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990. |

[23] | Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990. |

[24] | Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990. |

[25] | Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990. |

[26] | Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. |

[27] | Edmund Woronowicz. Many-argument relations. Formalized Mathematics, 1(4):733-737, 1990. |

[28] | Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990. |

[29] | Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990. |

[30] | Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990. |

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