Peirce on Perfect Sets, Revised

Peirce's definition of 'continuity' for the Century Dictionary, as reprinted in the Collected Papers, contains what appears to be a serious mathematical error: the auxiliary definition of perfect sets that Peirce attributes to Cantor is not just different from the latter's, but also inconsistent with other statements Peirce makes in his own dictionary entry. The inconsistency disappears when we revert to Peirce's original text; the Collected Papers version incorporates a revision which Peirce marked, and then canceled, in his copy of the dictionary. But even when that correction has been made, a slight divergence from Cantor's definition of perfect sets remains. I argue that Peirce's ongoing disregard of that divergence is bound up with some of the well-known differences between his standpoint and Cantor's, and with his protracted search for the completeness condition that distinguishes the continuum from the rational numbers.