This article argues against untyped pluralism by showing that it is subject to a variant of a Russell-style argument put forth by Timothy Williamson and that it clashes with a plural version of Cantor's theorem
. It concludes that pluralists should postulate a type distinction between objects and properties.
Peirce claimed prioricity on the diagonal proof of Cantor's Theorem
. Moore shows handily that Peirce's own proof of the theorem was, at the least, written several years after Cantor published his, and thus Peirce's claim to be the first to prove it was clearly false, though Moore argues it is at least possible that Peirce neither plagiarized Cantor's proof nor derived the theorem completely independently, but that, rather, some middle explanation is the correct one, e.g., that Peirce was inspired by Cantor's methodology and thought to derive the theorem he had not yet read.