 It seems very difficult to answer this question. It is not hard to see how a successor function and addition and multiplication operations can be defined on the number-candidates of I and on the number-candidates of II so that all the arithmetical statements that we take to be true come out true. Indeed, if this is done in the natural way, then we arrive at isomorphic structures (in the set-theoretic sense of the word), and isomorphic structures make the same sentences true (they are elementarily equivalent). It is only when we ask extra-arithmetical questions, such as ‘\(1 \in 3\)?’ that the two accounts of the natural numbers yield diverging answers. So it is impossible that both accounts are correct. According to story I, \(3 = \{\{\{\varnothing \}\}\}\), whereas according to story II, \(3 = \{\varnothing , \{\varnothing \}, \{\varnothing , \{\varnothing \}\}\}\). If both accounts were correct, then the transitivity of identity would yield a purely set theoretic falsehood.