Indiscernibles

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

${\displaystyle ([\phi (a,b)\wedge \phi (a,c)\wedge \phi (b,c)]\vee [\mathrm{\neg }\phi (a,b)\wedge \mathrm{\neg }\phi (a,c)\wedge \mathrm{\neg }\phi (b,c)])}$ and

${\displaystyle ([\phi (b,a)\wedge \phi (c,a)\wedge \phi (c,b)]\vee [\mathrm{\neg }\phi (b,a)\wedge \mathrm{\neg }\phi (c,a)\wedge \mathrm{\neg }\phi (c,b)]).}$

More generally, for a structure ${\displaystyle \mathfrak{A}}$ with domain ${\displaystyle A}$ and a linear ordering ${\displaystyle <}$, a set ${\displaystyle I\subseteq A}$ is said to be a set of ${\displaystyle <}$-indiscernibles for ${\displaystyle \mathfrak{A}}$ if for any finite subsets ${\displaystyle \{i_0,\dots ,i_n\}\subseteq I}$ and ${\displaystyle \{j_0,\dots ,j_n\}\subseteq I}$ with ${\displaystyle i_0<\dots <i_n}$ and ${\displaystyle j_0<\dots <j_n}$ and any first-order formula ${\displaystyle \varphi }$ of the language of ${\displaystyle \mathfrak{A}}$ with ${\displaystyle n}$ free variables, ${\displaystyle \mathfrak{A}\models \varphi (i_0,\dots ,i_n)⟺\mathfrak{A}\models \varphi (j_0,\dots ,j_n)}$.^{p. 2}

Applications

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