Download Elementi di Logica Matematica by Petr Novikov PDF

Download Elementi di Logica Matematica by Petr Novikov PDF

By Petr Novikov

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Example text

If the class % is characterizable by universal axioms, then the open formulas define the only formular predicates with this property. In a class of models which are algebras, however, this ceases to be the case, since all formulas representing equations of basic terms give predicates with the desired invariance. ,xn)be a FOPL formula representing a predicate invariant with respect to passage to sub- and superalgebras in a universally Derived operations and predicates 42 axiomatizable class % of algebras.

Stone's theorem on the representability of infinite boolean algebras serves as an example for this theorem. To see this, let % be a boolean algebra, take the predicate symbols Pa to be unary, and let the formulas correspond to the relations u = u+w and u = u' on 94. Representations of this type are then sought in the class defined by the ordinary FOPL sentences: where a, b are distinct elements ofA. 24 Representations of models As a second example we consider the so-called algebras of relations.

In the usual manner, instead of operations one can take the corresponding predicates and consider algebras to be models. Let us agree to say that a class % of algebras is ktcaZly definable (or just Zocaf) iff from the fact that every finite submodel of an arbitrary algebra % is isomorphic to a submodel of some %-algebra it follows that itself belongs to %. In case the basic operations of % are infinite in number, in this definition we take submodels relative to arbitrary finite subsystems of the basic operations.

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