Henryk Kotlarski (1949 – 2008) published over forty research articles, most of  them devoted to model theory of Peano arithmetic. He studied nonstandard satisfaction classes, automorphisms of models of Peano arithmetic, clasification of elementary cuts, ordinal combinatorics of finite sets in the style of Ketonen and Solovay, and independence results. Zofa Adamowicz was a colleague of Henryk Kotlarski for about forty years. They did not write a joint paper but they had a lot of discussions and inspired one another. She shared the main interests of Henryk, in partcular the interest in the incompleteness phenomenon and various proofs of the second Gödel incompleteness theorem. Teresa Bigorajska is a Ph D student of Zofia Adamowicz and a major collaborator of Henryk Kotlarski during his last years. They worked together on ordinal combinatorics of finite sets – a notion heavily used in the book. They studied combinatorial properties of partitions and trees with respect to the notion of largness in the style of Ketonen and Solovay. They developed the machinery for proving independence results presented in the book. Konrad Zdanowski »s research interests focus on theories of arithmetic,   intuitionistic logic, and philosophy. Konrad Zdanowski worked with Henryk Kotlarski on one of his last articles and, through many conversations, he learned from Henryk  some of his approach to arithmetic.