2

Why did they find it necessary to introduce the notion of an empty domain (and the related changes)? I.e. why wasn't assuming 'all' to be understood as 'all, if any' (and respectively, 'some' as 'some, if any') not satisfactory?

3

The empty domain or empty set was introduced for formal reasons only.

Georg Cantor mentioned the empty set with some reservations and only once in all his work: "Further it is useful to have a symbol expressing the absence of points. We choose for that sake the letter O. P = O means that the set P does not contain any single point. So it is, strictly speaking, not existing as such."

Bertrand Russell considered an empty class as not existing: "An existent class is a class having at least one member."

And even Ernst Zermelo who made the "ZFC-Axiom: There is an (improper) set, the 'null-set' O which does not contain any element" said in private correspondence: "It is not a genuine set and was introduced by me only for formal reasons."

W. Mückenheim: Transfinity - A Source Book.

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