Benacerraf's 1965 multiple-reductions argument depends on what I call 'deferential logicism': his necessary condition for number-set identity is most plausible against a background Quineanism that allows autonomy of the natural number concept. (This explains his neglect of 'theoretical-virtue' arguments.) Steinhart's 'folkist' sufficient condition on number-set identity, by contrast, puts that autonomy at the center - but fails for not taking the folk perspective seriously enough. Learning from both sides, we explore new conditions on number-set identity, elaborating a (1983) suggestion from Wright.
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