# Preliminaries to Classical First Order Model Theory

Formalized Mathematics (2011)

- Volume: 19, Issue: 3, page 155-167
- ISSN: 1426-2630

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topMarco Caminati. "Preliminaries to Classical First Order Model Theory." Formalized Mathematics 19.3 (2011): 155-167. <http://eudml.org/doc/267874>.

@article{MarcoCaminati2011,

abstract = {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 Lowenheim-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 (see [20], definition on page 5), 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.},

author = {Marco Caminati},

journal = {Formalized Mathematics},

keywords = {Mizar Mathematical Library},

language = {eng},

number = {3},

pages = {155-167},

title = {Preliminaries to Classical First Order Model Theory},

url = {http://eudml.org/doc/267874},

volume = {19},

year = {2011},

}

TY - JOUR

AU - Marco Caminati

TI - Preliminaries to Classical First Order Model Theory

JO - Formalized Mathematics

PY - 2011

VL - 19

IS - 3

SP - 155

EP - 167

AB - 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 Lowenheim-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 (see [20], definition on page 5), 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.

LA - eng

KW - Mizar Mathematical Library

UR - http://eudml.org/doc/267874

ER -

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## Citations in EuDML Documents

top- Marco B. Caminati, Artur Korniłowicz, Pseudo-Canonical Formulae are Classical
- Marco Caminati, Definition of First Order Language with Arbitrary Alphabet. Syntax of Terms, Atomic Formulas and their Subterms
- Marco Caminati, First Order Languages: Further Syntax and Semantics
- Karol Pąk, Flexary Operations
- Karol Pąk, Euler’s Partition Theorem
- Marco Caminati, Sequent Calculus, Derivability, Provability. Gödel's Completeness Theorem
- Marco Caminati, Free Interpretation, Quotient Interpretation and Substitution of a Letter with a Term for First Order Languages

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