ABSTRACT
This chapter talks about two important first-order language-systems of a much more powerful kind. These are a system based upon the Zermelo set theory, and one based upon Russell's simplified theory of types. The reasons for choosing the language-systems are threefold. First they are both extremely important language-systems which have attracted wide attention and have been before the learned world for some years. And secondly, a meta-language such as can be taken as the semantical meta-language for either of these kinds of systems, in the same uniform way as for simpler systems. A third reason for considering the powerful languages in detail rather than narrower languages is that most well-developed scientific or philosophical languages can be regarded as one or the other of the languages appropriately extended by the addition of further non-logical primitive constants. Most well-developed scientific or philosophical languages contain a good deal of mathematics.
