 Of course, the vicious circle principle is not itself a theory, but a condition any adequate theory has to satisfy. Russell (1906, 1907) tentatively proposed three alternative approaches: the zigzag theory, the theory of limitation of size, and the no-classes theory. The zigzag theory attempts to capture the idea that the right functions ought to be ‘simple’, while according to the second view, ‘predicative’ would be characterized by a certain limitation of the size of classes which can be predicatively defined (e.g., the collection of all ordinals \(\mathbf{ON}\) is too large). In the no-classes theory, classes are not independent entities and anything said about them is to be regarded as an abbreviation of a statement about their members and the propositional functions defining them. This is not far from Poincaré’s idea that mathematical objects should be specified in finitely many words. However, Russell developed his own technical devices—such as the ‘substitution method’ (Landini 1998), and the contextual elimination of definite descriptions (see the entry  Russell)  – in order to implement his ideas.