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Indiscernibles

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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).