In mathematical logic, indiscernibles are objects which cannot be distinguished by any property or relation defined by a formula. Usually only first-order formulas are considered. For example, if {A, B, C} is indiscernible, then for each 2-ary formula K, we must have K(A, B) ⇔ K(B, A) ⇔ K(C, A) ⇔ K(A, C) ⇔ K(B, C) ⇔ K(C, B).
It is also common to consider "order-indiscernibles", which possess a total ordering, and satisfy relations dependent only on the relative order of the arguments. If the set above were only order-indiscernible (and ordered alphabetically), we would have K(A, B) ⇔ K(A, C) ⇔ K(B, C), but it would not necessarily be true that K(A, B) ⇔ K(B, A).
Order-indiscernibles feature prominently in the theory of Ramsey cardinals, Erdős cardinals, and Zero sharp.
Historically, the identity of indiscernibles was one of the laws of thought of Gottfried Leibniz.
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