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 [alpha]-equivalence.<br />Our presentation uses a novel presentation based not on a theory but on a concrete model .
Murdoch J. Gabbay. (2007). A general mathematics of names. Information and Computation, 205(7), 982–1011. https://doi.org/10.1109/DYSPAN.2005.1542626