Theory (mathematical logic)

Lore & Background

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, the elements of which are called statements. These initial statements are often called the primitive elements or elementary statements of the theory. A theory is a conceptual class consisting of certain of these elementary statements; those that belong to the theory are called the elementary theorems and are said to be true. In this way, a theory can be seen as a way of designating a subset of the elementary statements that 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 the theory, so the same elementary statement may be true with respect to one theory but false with respect to another.

Reader's Guide

A theory is a foundational concept in mathematical logic, serving as a structured collection of sentences in a formal language. Its significance lies in its role as a framework for deductive reasoning: given a deductive system, a theory is closed under logical consequence, meaning that any sentence provable from the theory is itself a theorem of the theory. The notion of axioms—a subset of the theory—allows for the systematic generation of theorems via inference rules, forming an axiomatic system. Key properties of theories include consistency (syntactic or semantic) and completeness. A syntactically consistent theory is one from which not every sentence can be proven; a satisfiable theory has a model. For first-order logic, the completeness theorem ensures that syntactic consistency and satisfiability coincide, but in other logics (e.g., second-order logic) syntactically consistent theories may not be satisfiable. A complete consistent theory is one where for every sentence in its language, either the sentence or its negation is provable. The concept of subtheories and extensions allows for hierarchical relationships between theories. The legacy of this concept is its centrality to the foundations of mathematics, model theory, and proof theory, providing a precise language for discussing truth, provability, and the structure of mathematical knowledge.

Did You Know?

• A theory is a set of sentences in a formal language, and an element of a deductively closed theory is called a theorem.
• Every axiom is automatically a theorem by definition.
• A syntactically consistent theory is one from which not every sentence in the underlying language can be proven.
• For first-order logic, the completeness theorem implies that syntactic consistency and satisfiability coincide.