In mathematical logic, a definable set is an n-ary relation on the domain of a structure whose elements satisfy some formula in the first-order language of that structure. A set can be defined with or without parameters, which are elements of the domain that can be referenced in the formula defining the relation.

Definition

a fixed subset a natural number. Then:

• is ,

• The bracket notation here indicates the semantic evaluation of the free variables in the formula.

• A set without parameters with parameters from the empty set (that is, with no parameters in the defining formula).

• .

• (with parameters) if the singleton set (with those parameters).

Examples

The natural numbers with only the order relation

stating that there exist no elements less than x:

elements less than x:

In contrast, one cannot define any specific integer consisting of the integers with the usual ordering (see the section on automorphisms below).

The natural numbers with their arithmetical operations

be the first-order structure consisting of the natural numbers and their usual arithmetic operations and order relation. The sets definable in this structure are known as the arithmetical sets, and are classified in the arithmetical hierarchy. If the structure is considered in second-order logic instead of first-order logic, the definable sets of natural numbers in the resulting structure are classified in the analytical hierarchy. These hierarchies reveal many relationships between definability in this structure and computability theory, and are also of interest in descriptive set theory.

The field of real numbers

be the structure consisting of the field of real numbers. Although the usual ordering relation is not directly included in the structure, there is a formula that defines the set of nonnegative reals, since these are the only reals that possess square roots:

is called a definitional extension of the original structure. It has the same expressive power as the original structure, in the sense that a set is definable over the enlarged structure from a set of parameters if and only if it is definable over the original structure from that same set of parameters.

The theory has quantifier elimination. Thus the definable sets are Boolean combinationss of solutions to polynomial equalities and inequalities; these are called semi-algebraic sets. Generalizing this property of the real line leads to the study of o-minimality.

Invariance under automorphisms

An important result about definable sets is that they are preserved under automorphisms which fix their parameter set.

1. ,

itself. In contrast, there are infinitely many definable sets of pairs (or indeed n-tuples for any fixed n >: (in the case n. In particular, any automorphism (translation) preserves the "distance" between two elements.

Additional results

The Tarski–Vaught test is used to characterize the elementary substructures of a given structure.

References

• Hinman, Peter. Fundamentals of Mathematical Logic, A K Peters, 2005.

• Marker, David. Model Theory: An Introduction, Springer, 2002.

• Rudin, Walter. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill, 1976.

• Slaman, Theodore A. and Woodin, W. Hugh. Mathematical Logic: The Berkeley Undergraduate Course. Spring 2006.