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In
mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry) ...
, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios, a deductive system is first understood from context, after which an element \phi\in T of a theory T is then called a theorem of the theory. In many deductive systems there is usually a subset \Sigma \subset T that is called "the set of
axioms An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...

axioms
" of the theory
T
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, in which case the deductive system is also called an "
axiomatic system In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms.


General theories (as expressed in formal language)

When defining theories for foundational purposes, additional care must be taken as normal set-theoretic language may not be appropriate. The construction of a theory begins by specifying a definite non-empty ''conceptual class'' \mathcal, the elements of which are called ''statements''. These initial statements are often called the ''primitive elements'' or ''elementary'' statements of the theory — to distinguish them from other statements which may be derived from them. A theory \mathcal is a conceptual class consisting of certain of these elementary statements. The elementary statements which belong to \mathcal are called the ''elementary theorems'' of \mathcal and are said to be ''true''. In this way, a theory can be seen as a way of designating a subset of \mathcal which only contain statements that are true. This general way of designating a theory stipulates that the truth of any of its elementary statements is not known without reference to \mathcal. Thus the same elementary statement may be true with respect to one theory but false with respect to another. This is reminiscent of the case in ordinary language where statements such as "He is an honest person" cannot be judged true or false without interpreting who "he" is, and — for that matter — what an "honest person" is under this theory. Haskell Curry, ''Foundations of Mathematical Logic'', 2010.


Subtheories and extensions

A theory ''\mathcal'' is a subtheory of a theory ''\mathcal'' if ''\mathcal'' is a subset of ''\mathcal''. If ''\mathcal'' is a subset of ''\mathcal'' then ''\mathcal'' is called an extension or a supertheory of ''\mathcal''


Deductive theories

A theory is said to be a ''deductive theory'' if \mathcal is an inductive class. That is, that its content is based on some formal deductive system and that some of its elementary statements are taken as
axioms An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
. In a deductive theory, any sentence which is a
logical consequence Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more statements. A Validity (log ...
of one or more of the axioms is also a sentence of that theory.


Consistency and completeness

A syntactically consistent theory is a theory from which not every sentence in the underlying language can be proven (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 In classical logic, intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of ...
, this is equivalent to requiring that there is no sentence φ such that both φ and its negation can be proven from the theory. A satisfiable theory is a theory that has a model. This means there is a structure ''M'' that satisfiability, 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 Gödel's completeness theorem, 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 theory, complete consistent theory (or just a complete theory) is a consistent theory ''\mathcal'' such that for every sentence φ in its language, either φ is provable from ''\mathcal'' or ''\mathcal'' \cup is inconsistent. For theories closed under logical consequence, this means that for every sentence φ, either φ or its negation is contained in the theory. An incomplete theory is a consistent theory that is not complete. (see also ω-consistent theory for a stronger notion of consistency.)


Interpretation of a theory

An ''interpretation of a theory'' is the relationship between a theory and some subject matter when there is a many-to-one correspondence between certain elementary statements of the theory, and certain statements related to the subject matter. If every elementary statement in the theory has a correspondent it is called a ''full interpretation'', otherwise it is called a ''partial interpretation''. Here: p.48


Theories associated with a structure

Each Structure (mathematical logic), structure has several associated theories. The complete theory of a structure ''A'' is the set of all first-order sentences over the Signature (logic), signature of ''A'' which are satisfied by ''A''. It is denoted by Th(''A''). More generally, the theory of ''K'', a class of σ-structures, is the set of all first-order σ-sentences that are satisfied by all structures in ''K'', and is denoted by Th(''K''). Clearly Th(''A'') = Th(). These notions can also be defined with respect to other logics. For each σ-structure ''A'', there are several associated theories in a larger signature σ' that extends σ by adding one new constant symbol for each element of the domain of ''A''. (If the new constant symbols are identified with the elements of ''A'' which they represent, σ' can be taken to be σ \cup A.) The cardinality of σ' is thus the larger of the cardinality of σ and the cardinality of ''A''. The diagram of ''A'' 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 (model theory), expansion of ''A'' to the signature σ'.


First-order theories

A first-order theory \mathcal is a set of sentences in a first-order formal language \mathcal.


Derivation in a first-order theory

There are many formal derivation ("proof") systems for first-order logic. These include Hilbert-style deductive system, Hilbert-style deductive systems, natural deduction, the sequent calculus, the Method of analytic tableaux, tableaux method and Resolution (logic), resolution.


Syntactic consequence in a first-order theory

A well-formed formula, formula ''A'' is a syntactic consequence of a first-order theory \mathcal if there is a formal proof, derivation of ''A'' using only formulas in \mathcal as non-logical axioms. Such a formula ''A'' is also called a theorem of \mathcal. The notation " \mathcal \vdash A" indicates ''A'' is a theorem of \mathcal.


Interpretation of a first-order theory

An interpretation of a first-order theory provides a semantics for the formulas of the theory. An interpretation is said to satisfy a formula if the formula is true according to the interpretation. A model of a first-order theory \mathcal is an interpretation in which every formula of \mathcal is satisfied.


First-order theories with identity

A first-order theory \mathcal is a first-order theory with identity if \mathcal includes the identity relation symbol "=" and the reflexivity and substitution axiom schemes for this symbol.


Topics related to first-order theories

*Compactness theorem *Consistent set *Deduction theorem *Enumeration theorem *Lindenbaum's lemma *Löwenheim–Skolem theorem


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 (mathematical logic), structure, and let the theory to be the set of sentences that are satisfied by the structure. This is a method for producing complete theories through the semantic route, with examples including the set of true sentences under the structure (N, +, ×, 0, 1, =), where N is the set of natural numbers, and the set of true sentences under the structure (R, +, ×, 0, 1, =), where R is the set of real numbers. The first of these, called the theory of true arithmetic, cannot 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 Decidability (logic), decidable; it is the theory of real closed fields (see Decidability of first-order theories of the real numbers for more).


See also

* Axiomatic system * Interpretability * List of first-order theories *Mathematical theory


References


Further reading

* {{Mathematical logic Formal theories, Logical expressions fr:Théorie axiomatique