Everything about Theory Mathematical Logic totally explained
In
mathematical logic, a
theory is a set of
sentences in a
formal language. E.g. a
first-order theory is a set of
first-order sentences. Many authors require that the theory be closed under
logical consequence.
Examples
One way to specify a theory is to define a set of
axioms in a particular language. The theory can be taken to include just those axioms, or their logical or provable consequences, as desired. Theories obtained this way include
ZFC and
Peano arithmetic.
A second way to specify a theory is to begin with a
structure and then let the theory be the set of formulas that are satisfied by the structure. This is one method for producing complete theories, described below. Examples of theories of this sort include the sets of true sentences in the structures (
N, +, ×, 0, 1, =) and (
R, +, ×, 0, 1, =), where
N is the set of natural numbers and
R is the set of real numbers. The first of these, called the theory of true arithmetic, can't be written as the set of logical consequences of any
enumerable set of axioms.
The theory of (
R, +, ×, 0, 1, =) was shown by Tarski to be
decidable; it's the theory of
real closed fields.
Consistency and completeness
A
syntactically consistent theory is a theory from which not every sentence in the underlying language can be proved (with respect to some
deductive system which is usually clear from context). In a deductive system (such as first-order logic) that satisfies the
principle of explosion, this is equivalent to requiring that there's no sentence φ such that both φ and its negation can be proved from the theory.
A
satisfiable theory is a theory that has a model. This means there's a structure
M that satisfies every sentence in the theory. Any satisfiable theory is syntactically consistent, because the structure satisfying the theory will satisfy exactly one of φ and the negation of φ, for each sentence φ.
A
consistent theory is sometimes defined to be a syntactically consistent theory, and sometimes defined to be a satisfiable theory. For
first-order logic, the most important case, it follows from the
completeness theorem that the two meanings coincide. In other logics, such as
second-order logic, there are syntactically consistent theories that are not satisfiable, such as
ω-inconsistent theories.
A
complete consistent theory (or just a
complete theory) is a
consistent theory
T such that for every sentence φ in its language, either φ is provable from
T or
T ). These notions can also be defined with respect to other logics.
The
diagram of a σ-structure
A is a theory in the signature σ' = σ
A, for example σ together with one new constant symbol each for every element of the domain of
A. (The new constant symbols are identified with the elements of
A which they represent.) The theory consists of all atomic or negated atomic σ-sentences that are satisfied by
A and is denoted by diag
A. The
positive diagram of
A is the set of all atomic σ-sentences which
A satisfies. It is denoted by diag
+A. The
elementary diagram of
A is the set eldiag
A of
all first-order σ'-sentences that are satisfied by
A or, equivalently, the complete (first-order) theory of the natural
expansion of
A to the signature σ'.
Further Information
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