In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, indiscernibles are objects which cannot be distinguished by any property

Property (philosophy)

In modern philosophy, logic, and mathematics a property is an attribute of an object; a red object is said to have the property of redness. The property may be considered a form of object in its own right, able to possess other properties. A property however differs from individual objects in that...

 or relation

Relation (mathematics)

In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...

 defined by a formula

Well-formed formula

In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...

. Usually only first-order

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

 formulas are considered. For example, if {a, b, c} is a set of indiscernibles, then for each 2-ary formula φ, we must have φ(a,b) ⇔

IFF

IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...

 φ(b,a) ⇔ φ(c,a) ⇔ φ(a,c) ⇔ φ(b,c) ⇔ φ(c,b).
Historically, the identity of indiscernibles

Identity of indiscernibles

The identity of indiscernibles is an ontological principle which states that two or more objects or entities are identical if they have all their properties in common. That is, entities x and y are identical if any predicate possessed by x is also possessed by y and vice versa...

 was one of the laws of thought of Gottfried Leibniz

Gottfried Leibniz

Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

.

In some contexts one considers the more general notion of order-indiscernibles, and the term sequence of indiscernibles often refers implicitly to this weaker notion.
For example, to say that the triple (a, b, c) is a sequence of indiscernibles amounts to saying φ(a,b) ⇔ φ(a,c) ⇔ φ(b,c), but leaves open whether φ(a,b) ⇔ φ(b,a) holds or not.

Order-indiscernibles feature prominently in the theory of Ramsey cardinal

Ramsey cardinal

In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by and named after Frank P. Ramsey.With [κ]<ω denoting the set of all finite subsets of κ, a cardinal number κ such that for every function...

s, Erdős cardinal

Erdos cardinal

In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by .The Erdős cardinal κ is defined to be the least cardinal such that for every function...

s, and Zero sharp

Zero sharp

In the mathematical discipline of set theory, 0# is the set of true formulas about indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers , or as a subset of the hereditarily finite sets, or as a real number...

.