Логикалық теориялардың модельдерінің болуы мен кардиналдығы

Existence and cardinality of models of logical theories
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first order theories are unable to control the cardinality of their infinite models. The (downward) Löwenheim–Skolem theorem is one of the two key properties, along with the compactness theorem, that are used in Lindström's theorem to characterize first order logic. In general, the Löwenheim–Skolem theorem does not hold in stronger logics such as second order logic.

Below we elaborate on the general concept of signatures and structures.

A signature consists of a set of function symbols Sfunc, a set of relation symbols Srel, and a function representing the arity of function and relation symbols. (A nullary function symbol is called a constant symbol.) In the context of first order logic, a signature is sometimes called a language. It is called countable if the set of function and relation symbols in it is countable, and in general the cardinality of a signature is the cardinality of the set of all the symbols it contains. A first order theory consists of a fixed signature and a fixed set of sentences (formulas with no free variables) in that signature. Theories are often specified by giving a list of axioms that generate the theory, or by giving a structure and taking the theory to consist of the sentences satisfied by the structure.

The statement given in the introduction follows immediately by taking M to be an infinite model of the theory. The proof of the upward part of the theorem also shows that a theory with arbitrarily large finite models must have an infinite model; sometimes this is considered to be part of the theorem.

First, one extends the signature by adding a new constant symbol for every element of M. The complete theory of M for the extended signature σ' is called the elementary diagram of M. In the next step one adds κ many new constant symbols to the signature and adds to the elementary diagram of M the sentences c ≠ c' for any two distinct new constant symbols c and c'. Using the compactness theorem, the resulting theory is easily seen to be consistent. Since its models must have cardinality at least κ, the downward part of this theorem guarantees the existence of a model N which has cardinality exactly κ. It contains an isomorphic copy of M as an elementary substructure.

Although the (classical) Löwenheim–Skolem theorem is tied very closely to first order logic, variants hold for other logics. For example, every consistent theory in second order logic has a model smaller than the first supercompact cardinal (assuming one exists). The minimum size at which a (downward) Löwenheim–Skolem–type theorem applies in a logic is known as the Löwenheim number, and can be used to characterize that logic's strength. Moreover, if we go beyond first order logic, we must give up one of three things: countable compactness, the downward Löwenheim–Skolem Theorem, or the properties of an abstract logic.