Equipe(s) : | lm, |

Responsables : | S. Anscombe, A. Vignati |

Email des responsables : | sylvy.anscombe@imj-prg.fr, vignati@imj-prg.fr |

Salle : | 1013 |

Adresse : | Sophie Germain |

Description |

Orateur(s) | Joel Hamkins - University of Oxford, |

Titre | Determinacy for proper class games |

Date | 14/04/2021 |

Horaire | 16:00 à 17:15 |

Diffusion | |

Résume | The principle of open determinacy for class games —
two-player games of perfect information with plays of length ω, where
the moves are chosen from a possibly proper class, such as games on
the ordinals — is not provable in Zermelo-Fraenkel set theory ZFC or
Gödel-Bernays set theory GBC, if these theories are consistent,
because provably in ZFC there is a definable open proper class game
with no definable winning strategy. In fact, the principle of open
determinacy and even merely clopen determinacy for class games implies
Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it
implies that there is a satisfaction class for first-order truth, and
indeed a transfinite tower of truth predicates for iterated
truth-about-truth, relative to any class parameter. This is perhaps
explained, in light of the Tarskian recursive definition of truth, by
the more general fact that the principle of clopen determinacy is
exactly equivalent over GBC to the principle of elementary transfinite
recursion ETR over well-founded class relations. Meanwhile, the
principle of open determinacy for class games is strictly stronger,
although it is provable in the stronger theory GBC+
Pi^1_1-comprehension, a proper fragment of Kelley-Morse set theory KM. http://jdh.hamkins.org/determinacy-for-proper-class-games-seminaire-de-logique-lyon-paris-april-2021/ |

Salle | 1013 |

Adresse | Sophie Germain |

© IMJ-PRG