Forcing, Downward Löwenheim-Skolem and Omitting Types Theorems, Institutionally

• Published in: Logica universalis Vol. 8; no. 3-4; pp. 469 - 498
• Main Author: Găină, Daniel
• Format: Journal Article
• Language: English
• Published: Basel Springer Basel 01.12.2014
• Subjects: Computer Science; Logic; Mathematics; Mathematics and Statistics; Primary 03C25; Secondary 03C95; model theory; forcing; Institution
• ISSN: 1661-8297; 1661-8300
• DOI: 10.1007/s11787-013-0090-0

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 two abstract results: Downward Löwenheim-Skolem Theorem (DLST) and Omitting Types Theorem (OTT). We instantiate these general results to many first-order logics, which are, roughly speaking, logics whose sentences can be constructed from atomic formulas by means of Boolean connectives and classical first-order quantifiers. These include first-order logic ( FOL ), logic of order-sorted algebras ( OSA ), preorder algebras ( POA ), as well as their infinitary variants FOL ω 1 , ω , OSA ω 1 , ω , POA ω 1 , ω . In addition to the first technique for proving OTT, we develop another one, in the spirit of institution-independent 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 partial algebras ( PA ) and higher-order logic with Henkin semantics ( HNK ), and from the institution of FOL ω 1 , ω to PA ω 1 , ω and HNK ω 1 , ω . The second technique successfully extends the domain of application of OTT to non conventional logical systems for which the standard methods may fail.