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

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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

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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

  and
 

More generally, for a structure   with domain   and a linear ordering  , a set   is said to be a set of  -indiscernibles for   if for any finite subsets   and   with   and   and any first-order formula   of the language of   with   free variables,  .[1]p. 2

Applications

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

See also

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References

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  • 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.

Citations

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  1. ^ J. Baumgartner, F. Galvin, "Generalized Erdős cardinals and 0#". Annals of Mathematical Logic vol. 15, iss. 3 (1978).