Translating the hypergame paradox: Remarks on the set of founded elements of a relation
type: Item
instance of: scholarly article
title: Translating the hypergame paradox: Remarks on the set of founded elements of a relation
zbMATH Open document ID: 0859.03004
DOI: 10.1007/BF00257385
author: Claudio Bernardi
author: Giovanna D'Agostino
published in: Journal of Philosophical Logic
publication date: 1996-11-12T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 03A05
Mathematics Subject Classification ID: 03B30
Mathematics Subject Classification ID: 91A05
zbMATH DE Number: 944232
zbMATH Keywords: Peano arithmetic
zbMATH Keywords: founded game
zbMATH Keywords: unfounded game
zbMATH Keywords: hypergame paradox
zbMATH Keywords: diagonal arguments
zbMATH Keywords: Burali-Forti's paradox
zbMATH Keywords: founded elements in an r.e. relation
zbMATH Keywords: diagonalisable algebras
zbMATH Keywords: undecidable formulas in ZF set theory
zbMATH Keywords: Kripke's theory of truth
reviewed by: Jim Mackenzie
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00257385
OpenAlex ID: W1989037937
description: scientific article; zbMATH DE number 944232