A general mathematics of names

We introduce FMG (Fraenkel–Mostowski Generalised) set theory, a generalisation of FM set theory which allows binding of infinitely many names instead of just finitely many names. We apply this generalisation to show how three presentations of syntax—de Bruijn indices, FM sets, and name-carrying syntax—have a relation generalising to all sets and not only sets of syntax trees. We also give syntax-free accounts of Barendregt representatives, scope extrusion, and other phenomena associated to α-equivalence.Our presentation uses a novel presentation based not on a theory but on a concrete model U.