Forcing and the Omitting Type Theorem, institutionally
نویسنده
چکیده
In the context of proliferation of many logical systems in the area of mathematical logic and computer science, we present a generalization of forcing in institution-independent model theory which is used to prove an abstract Omitting Types Theorem (OTT). We instantiate this general result to many first-order logics, which are, roughly speaking, logics whose sentences can be constructed from atomic formulae by means of Boolean connectives and classical first-order quantifiers. These include first-order logic (FOL), logic of order-sorted algebra (OSA), preorder algebra (POA), partial algebras (PA), as well as their infinitary variants FOLω1,ω, OSAω1,ω, POAω1,ω, PAω1,ω. In addition to the first technique for proving the OTT, we develop another one, in the spirit of institutionindependent model theory, which consists of borrowing the Omitting Types Property (OTP) from a simpler institution across an institution comorphism. As a result we export the OTP from FOL to first-order partial algebras (FOPA) and higher-order logic with Henkin semantics (HNK), and from the institution of FOLω1,ω to FOPAω1,ω and HNKω1,ω. The second technique successfully extends the domain of application of OTT to (non classical) logical systems for which the standard methods may fail.
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