Indiscernibles

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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 a set of indiscernibles, then for each 2-ary formula φ, we must have φ(a,b) φ(b,a) ⇔ φ(c,a) ⇔ φ(a,c) ⇔ φ(b,c) ⇔ φ(c,b). Historically, the identity of indiscernibles was one of the laws of thought of Gottfried Leibniz.

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 cardinals, Erdős cardinals, and Zero sharp.

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