# What were the inconsistent classes identified by Ernst Schröder?

From G.H.Moore’s Zermelo’s Axiom of Choice :

In 1890 Schröder had already introduced a notion of consistent and inconsistent classes in his Algebra der Logik, well before the modern discovery of paradoxes in set theory and logic, and had utilized such inconsistent classes under certain conditions.

1890 is seven years before Cantor’s first published treatment of inconsistent classes in the second installment of Beiträge.

Since Schröder is working with and extending the methods of algebraic logic developed by Boole, I am guessing that these inconsistent classes are collections such as “the collection of all thoughts”, or somesuch.

There is an original German language PDF of Schröder’s Algebra of Logik online at archive.org, however a search inside this PDF for inkonsistent does not produce any matches.

Q: What were Schröder’s inconsistent classes and how did he demonstrate their inconsistency?

See :

Schroeder's argument involves the universe of discourse and is similar to the argument regarding the universal class.

EDIT

Here is the argument as recounted by Frege using Boole's 0, 1 notation:

As we have laid down, 0 would have to be contained in every class that can be got out of the manifold 1; … 0 would have to be the subject of every predicate. Now suppose we took a to be the class of those classes of the manifold that are equal to 1 (which would certainly be permissible if we could bring everything thinkable into the manifold 1), then this class of its very nature contains just one class, viz. the symbol 1 itself, or alternatively the whole of the manifold, which constitutes the reference of the symbol; but therefore besides this it would contain “nothing,” i.e. 0. Hence 1 and 0 would make up the class of the objects that are to be equal to 1; and so we should have to admit not only: 1 = 1 but also: 0 = 1. For a predicate that applies to a class -- in our case, the predicate: to be identically equal to 1 -- must also apply to every individual in the class, by Principle II. (Frege, 1895, p.91)

• Thanks! This paper does indeed contain Schröder’s argument, as recounted by Frege (page 12 of the linked PDF). I’ve edited your answer to include Frege’s reading.
– nwr
Commented Nov 8, 2016 at 19:48