Models with second order properties V: A general principle

Shelah, S., C. Laflamme and B. Hart, Models with second order properties V: A general principle, Annals of Pure and Applied Logic 64 (1993) 169-194. We present a general framework for carrying out the construction in [2-10] and others of the same type. The unifying factor is a combinatorial principle which we present in terms of a game in which the first player challenges the second player to carry out constructions which would be much easier in a generic extension of the universe, and the second player cheats with the aid of {black diamond suit}. Section 1 contains an axiomatic framework suitable for the description of a number of related constructions, and the statement of the main theorem 1.9 in terms of this framework. In Section 2 we illustrate the use of our combinatorial principle. The proof of the main result is then carried out in Sections 3-5.