Theory (mathematical logic)

In mathematical logic, a theory is ( the first-order predicate logic) a set of statements about a signature.

• 4.1 The theory of natural numbers
• 4.2 The theory of dense linear order without endpoints
• 4.3 The theory of algebraically closed field ( of characteristic p or 0)

Definition

A set of propositions is called deductively closed if for all statements

Already follows that

When a language is, a theory is a deductive abgeschlosse set of statements about this language.

(Note: The definitions in the literature are not uniform Partly it also does not require that a theory is deductively closed. . )

A set of statements is a axioms of a theory, if the deductive conclusion of these statements is the theory.

If a structure is, it is a theory, since this quantity is deductively closed.

Properties

Generally

• The cardinality of a theory is its cardinality as a set, but at least countable.
• A theory is consistent if it does not contain any rate. (This is equivalent thereto, that it " does not contain a set of the form." )
• The theory is complete if it contains either it or its negation of each statement.
• The theory is finally axiomatizable if it is the deductive conclusion of a finite set of statements.

Model Theory

• One theory is model complete if the fact that a model is in the other, this also is elementary then in the other.
• One theory has quantifier elimination if it is of the deductive conclusion of a set of formulas, which was formed without quantifiers.
• One theory is categorical in a cardinal number, if it has only one model of cardinality up to isomorphism.
• A complete theory is called small or narrow, if is countable for all. ( Is the set of all complete types in variables. )

Sets

Important sets of theories are:

Gödel completeness theorem:

• Every consistent theory has a model.

The Löwenheim - Skolem:

• If a theory has a model in an infinite cardinal number, it also has one in each cardinal number greater than or equal to their thickness.

The set of Morley:

• Is a countable theory in an uncountable cardinal number categorically, so in each.

Examples

The theory of natural numbers

The theory of the natural numbers is formulated on the language, the axioms formalize the following statements:

• Null is not a value of the ( successor function ) S.
• The successor function is injective
• For all
• For all
• For all
• For all
• For all

Additionally there is any formula for the induction formula with an axiom:

The theory of the natural numbers is incomplete. There is no consistent recursively enumerable expansion of natural numbers. This is the statement of the incompleteness theorem.

The theory of dense linear order without endpoints

The theory of dense linear order without endpoints is the theory of the rational numbers with the order relation "<". The axioms are to:

She has among others the following properties

• Finally, it is axiomatizable, but has no finite models.
• It is complete and model complete.
• All countable models are isomorphic (for evidence ), in uncountable cardinal numbers there are not isomorphic models. In the language of model theory that is: it is - categorical, but not categorical in uncountable cardinals: Is an uncountable cardinal number, so this theory has non- isomorphic models of cardinality.
• It is the (uniquely determined ) model companion of the theory of linear order.
• It has the rational numbers a prime model. (This is a model that can be embedded in any other elementary model. )
• Each model is atomic.
• She has quantifier elimination.
• It is not stable.

The theory of algebraically closed field ( of characteristic p or 0)

• The theory of algebraically closed fields without specifying the characteristic is model complete, but not completely.
• For the theory of algebraically closed fields with an indication of the characteristics apply:

• Is complete.
• She has a prime model.
• She is - categorical, but not categorically in an uncountable cardinal number.
• She has Quantorenelemination.