Concept in mathematical logic
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
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
![{\displaystyle [\beta (a,b)\land \beta (b,a)\land \beta (a,c)\land \beta (c,a)\land \beta (b,c)\land \beta (c,b)]\lor [\lnot \beta (a,b)\land \lnot \beta (b,a)\land \lnot \beta (a,c)\land \lnot \beta (c,a)\land \lnot \beta (b,c)\land \lnot \beta (c,b)]\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9101d86524adbf5fab4acfbc0e8805e11e4d63ce)
Historically, the
Gottfried Leibniz
.
Generalizations
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
![{\displaystyle ([\varphi (a,b)\land \varphi (a,c)\land \varphi (b,c)]\lor [\lnot \varphi (a,b)\land \lnot \varphi (a,c)\land \lnot \varphi (b,c)])\land ([\varphi (b,a)\land \varphi (c,a)\land \varphi (c,b)]\lor [\lnot \varphi (b,a)\land \lnot \varphi (c,a)\land \lnot \varphi (c,b)])\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33a88e315cab5dc8f787e0f333f7992db53cd911)
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
Order-indiscernibles feature prominently in the theory of Ramsey cardinals, Erdős cardinals, and zero sharp.
See also
References
