= Finite support in a permutation action
{title2=$G_{(S)}\subseteq\operatorname{Stab}(x),\quad |S|<\infty$}
= Finite support
{synonym}
Suppose a permutation group $G$ acts on an underlying atom set $A$ and on a collection of objects built over it. A finite subset $S\subseteq A$ supports an object $x$ when every permutation fixing all members of $S$ fixes $x$. Pairing supported objects uses the union of their supports. A definable subset or functional image inherits a support from its domain and parameters. In a <hereditarily finite-supported Quine-atom model>, every member is recursively required to have such a support; symmetry then rules out a choice function on the pairs of atoms.
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