Breaking the Tie: Benacerraf's Identification Argument Revisited.

The article challenges the philosophical consensus, originating from Benacerraf's argument in "What numbers could not be," that numbers cannot be reduced to sets due to the problem of multiple, indistinguishable set-theoretic reductions of arithmetic. It contests the widely accepted premise that there are no metaphysically relevant reasons to prefer one set-theoretic reduction over another, arguing instead that von Neumann ordinals are superior to alternatives like Zermelo ordinals based on mathematical practice, explanatory power, definitional simplicity, and foundational strength. The authors further address objections claiming these reasons are metaphysically irrelevant by showing that such features can be grounded in the concept of number itself, thus undermining Benacerraf's anti-reductionist argument. Consequently, the article supports set-theoretic reductionism by providing principled reasons to single out von Neumann ordinals as the correct set-theoretic representation of natural numbers.