Theory (logic)
In mathematical logic , a theory ( first-order predicate logic ) is a set of statements about a signature .
definition
A set of statements is called deductively closed if all statements are made
it already follows that
If there is a language, then a theory is a deductively closed set of statements about this language.
(Note: the definitions are not uniform in the literature. In some cases, it is also not required that a theory be deductively closed.)
A set of statements is a set of axioms for a theory if the deductive conclusion of those statements is the theory.
If there is a structure, it is a theory, since this set is deductively closed.
properties
General
- The power of a theory is its power as a set, or at least countable.
- A theory is consistent when it doesn't contain every sentence. (This is equivalent to not including a sentence of the form " ".)
- The theory is complete when it contains either it or its negation for each proposition.
- The theory is finitely axiomatizable if it is the deductive conclusion of a finite set of statements.
Model theory
- A theory is model-complete if the fact that one model is in the other means that it is also fundamentally in the other.
- A theory has quantifier elimination if it is the deductive closure of a set of formulas that were formed without quantifiers.
- A theory is categorical in a cardinal number if it has only one model of power apart from isomorphism .
- A complete theory is called small or narrow if it is countable for all . ( is the set of all complete types in variables.)
sentences
Important sentences about theories are:
The 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 every cardinal number greater than or equal to its power.
The set of Morley :
- If a countable theory is categorical in an uncountable cardinal number, then in every one.
Examples
The theory of natural numbers
The theory of natural numbers is formulated using language , the axioms formalize the following statements:
- Zero is not a value of the (successor function) S.
- The successor function is injective
- For everyone is
- For everyone is
- For everyone is
- For everyone is
- For everyone is
In addition, the induction formula with an axiom is for each formula :
- ( stands for )
The theory of natural numbers is incomplete. There is no consistent recursively enumerable expansion of the natural numbers. This is the statement of the incompleteness theorem .
The theory of dense linear order with no endpoints
The theory of the dense linear order without end points is the theory of the rational numbers with the order relation "<". The axioms are in detail:
- (Trichotomy)
- (Asymmetry)
- (Transitivity)
- (Openness)
- (Tightness)
It has the following properties, among others
- It is finitely axiomatizable, but has no finite models.
- It is complete and model-complete .
- All countable models are isomorphic (for proof ), in uncountable cardinal numbers there are non-isomorphic models. In the language of model theory this means: It is - categorical , but not categorical in uncountable cardinal numbers: If an uncountable cardinal number, then this theory has non-isomorphic models of power .
- It is the (uniquely determined) model companion of the theory of the linear order.
- With the rational numbers it has a prime model . (This is a model that can be fundamentally embedded in any other model.)
- Every model is atomic .
- It has quantifier elimination .
- It is not stable .
The theory of algebraically closed bodies (in the characteristic p or 0)
- The theory of algebraically closed bodies without specifying the characteristic is model-complete, but not complete.
- For the theory of algebraically closed bodies with an indication of the characteristic, the following applies:
- It is complete.
- She has a prime model.
- It is -categorical, but not categorical in an uncountable cardinal number.
- It has quantifier elimination.
Individual evidence
- ↑ Chang, Chen C., Keisler, H. Jerome: Model Theory. Amsterdam [u. a.], North-Holland, 1998, p. 12
literature
- H.-D. Ebbinghaus, J. Flum, W. Thomas: Introduction to Mathematical Logic , Spectrum Academic Publishing House, ISBN 3-8274-0130-5
- Wilfrid Hodges : Model theory. Cambridge University Press, 1993, ISBN 0-521-30442-3 .
- Chang, Chen C., Keisler, H. Jerome: Model Theory. Amsterdam [u. a.], North-Holland, 1998.
- Prestel, Alexander: Introduction to Mathematical Logic and Model Theory. Vieweg, Braunschweig 1986. (Vieweg course; 60: advanced course in mathematics). ISBN 3-528-07260-1 . 286 pp.
- Philipp Rothmaler: Introduction to Model Theory. Spektrum Akademischer Verlag, 1995, ISBN 978-3-86025-461-5 .