If a , b , and c are distinct and {a , b , c } is a set of indiscernibles , then, for example, for each binary formula
β
{\displaystyle \beta }
, we must have
[
β
(
a
,
b
)
∧
β
(
b
,
a
)
∧
β
(
a
,
c
)
∧
β
(
c
,
a
)
∧
β
(
b
,
c
)
∧
β
(
c
,
b
)
]
∨
[
¬
β
(
a
,
b
)
∧
¬
β
(
b
,
a
)
∧
¬
β
(
a
,
c
)
∧
¬
β
(
c
,
a
)
∧
¬
β
(
b
,
c
)
∧
¬
β
(
c
,
b
)
]
.
{\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)]\,.}
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. In our example of binary formulas, to say that the triple (a , b , c ) of distinct elements is a sequence of indiscernibles implies
(
[
φ
(
a
,
b
)
∧
φ
(
a
,
c
)
∧
φ
(
b
,
c
)
]
∨
[
¬
φ
(
a
,
b
)
∧
¬
φ
(
a
,
c
)
∧
¬
φ
(
b
,
c
)
]
)
∧
(
[
φ
(
b
,
a
)
∧
φ
(
c
,
a
)
∧
φ
(
c
,
b
)
]
∨
[
¬
φ
(
b
,
a
)
∧
¬
φ
(
c
,
a
)
∧
¬
φ
(
c
,
b
)
]
)
.
{\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)])\,.}
More generally, for a structure
A
{\displaystyle {\mathfrak {A}}}
with domain
A
{\displaystyle A}
and a linear ordering
<
{\displaystyle <}
, a set
I
⊆
A
{\displaystyle I\subseteq A}
is said to be a set of
<
{\displaystyle <}
-indiscernibles for
A
{\displaystyle {\mathfrak {A}}}
if for any finite subsets
{
i
0
,
…
,
i
n
}
⊆
I
{\displaystyle \{i_{0},\ldots ,i_{n}\}\subseteq I}
and
{
j
0
,
…
,
j
n
}
⊆
I
{\displaystyle \{j_{0},\ldots ,j_{n}\}\subseteq I}
with
i
0
<
…
<
i
n
{\displaystyle i_{0}<\ldots <i_{n}}
and
j
0
<
…
<
j
n
{\displaystyle j_{0}<\ldots <j_{n}}
and any first-order formula
ϕ
{\displaystyle \phi }
of the language of
A
{\displaystyle {\mathfrak {A}}}
with
n
{\displaystyle n}
free variables,
A
⊨
ϕ
(
i
0
,
…
,
i
0
)
⟺
A
⊨
ϕ
(
j
0
,
…
,
j
n
)
{\displaystyle {\mathfrak {A}}\vDash \phi (i_{0},\ldots ,i_{0})\iff {\mathfrak {A}}\vDash \phi (j_{0},\ldots ,j_{n})}
.[1 ] p. 2