# Indiscernibles

In mathematical logic, **indiscernibles** are objects that cannot be distinguished by any property or relation defined by a formula. Usually only first-order formulas are considered.

## Examples

If *a*, *b*, and *c* are distinct and {*a*, *b*, *c*} is a **set of indiscernibles**, then, for example, for each binary formula , we must have

Historically, the identity of indiscernibles was one of the laws of thought of Gottfried Leibniz.

## Generalizations

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. In our example of binary formulas, to say that the triple (*a*, *b*, *c*) of distinct elements is a sequence of indiscernibles implies

## Applications

Order-indiscernibles feature prominently in the theory of Ramsey cardinals, Erdős cardinals, and Zero sharp.

## See also

## References

- Jech, Thomas (2003).
*Set Theory*. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002.