Indiscernibles

Indiscernibles

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) if and only if K(B, A) if and only if K(C, A) if and only if K(A, C) if and only if K(B, C) if and only if 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) if and only if K(A, C) if and only if K(B, C), but not K(A, B) if and only if 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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