Benacerraf infers that any attempt to make such a choice of reduction immediately results in generating a meta-level, set-theoretic falsehood, namely in relation to other elementarily-equivalent set-theories not identical to the one chosen.
[1] Either method I or II could be used to define the natural numbers and subsequently generate true arithmetical statements to form a mathematical system.
[1] By attempting to reduce the natural numbers to sets, this renders a set-theoretic falsehood between the isomorphic structures of different mathematical systems.
He also demonstrates this with a story which Penelope Maddy describes in Realism in Mathematics: "Ernie and Johnny are both brought up on set theory.
His teachers define the operations of addition and multiplication on these sets, and when all the relabelling is done, Ernie counts and does arithmetic just like his schoolmates.
Johnny not only fails to share Ernie's enthusiasm, he declares the prized theorem to be outright false!
"[5] According to Benacerraf, the philosophical ramifications of this identification problem result in Platonic approaches failing the ontological test.
[1] The argument is used to demonstrate the impossibility for Platonism to reduce numbers to sets and reveal the existence of abstract objects.