definabilitya

Definability refers to the concept within logic and mathematics concerning whether a particular object, relation, or property can be precisely described using the formal language of a given theory. An object is said to be definable if there exists a formula within the language that uniquely identifies that object. Similarly, a relation or property is definable if there is a formula that precisely characterizes all and only those elements satisfying that relation or property.

The notion of definability is fundamental in model theory, set theory, and computability theory. It allows mathematicians to distinguish between objects that are explicitly constructed or characterized and those that might exist but cannot be explicitly specified. For instance, in Peano arithmetic, the number '0' is definable by a specific formula, while some non-standard models of arithmetic may contain elements that are not definable within the standard language of arithmetic.

Different notions of definability exist, such as definability with parameters, existential definability, and uniform definability. The strength and expressiveness of a formal language are often measured by what sorts of objects, relations, or properties are definable within it. The study of definability helps to understand the expressive power of logical systems and the structure of mathematical objects.