Translating the hypergame paradox: Remarks on the set of founded elements of a relation
A two-player game \(G\) is founded iff every contest of \(G\) must terminate after finitely many moves. A hypergame allows the first player to choose a founded game \(G\) and then the other player makes the first move of \(G\). Since \(G\) is founded, the hypergame is founded. If the second player also chooses hypergame, and so does the first player at the next turn, and so on, we have an unfounded game in which each player moves according to the rules of a founded game. Thus hypergame is founded, and if it is founded then it is not founded [\textit{W. S. Zwicker}, ``Playing games with games: the hypergame paradox'', Am. Math. Mon. 94, 507-514 (1987)]. This paper compares the hypergame argument with diagonal arguments, Burali-Forti's paradox, the complexity of the set of founded elements in an r.e. relation on the set of natural numbers, the theory of diagonalisable algebras, and the construction of undecidable formulas in ZF set theory and (with more difficulty) Peano Arithmetic, and with groundedness in Kripke's theory of truth [\textit{S. Kripke}, ``Outline of a theory of truth'', J. Philos. 72, 690-716 (1975)].
scientific article; zbMATH DE number 944232