Generic large cardinals: New axioms for mathematics?
type: Item
instance of: scholarly article
title: Generic large cardinals: New axioms for mathematics?
zbMATH Open document ID: 0917.03022
author: Matthew Foreman
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/231091
full work available at URL: Foreman.MAN.html
review text: This expository article surveys some prominent independence problems and discusses the various axioms (or axiom systems) which have been introduced to settle them: the axiom of constructibility, large cardinal hypotheses, AD, Martin's maximum, the generic large cardinal axioms. It focuses on the latter which assert the existence of large cardinal type elementary embeddings definable in forcing extensions of the universe. The advantage of generic elementary embeddings is that their critical point can be a small cardinal such as \(\omega_1\). Generic large cardinal axioms can often be reformulated as asserting the existence of ideals with strong properties, e.g., \(\kappa^+\)-saturated ideals on a regular cardinal \(\kappa\). The paper is an excellent introduction to the subject.
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03-02
zbMATH DE Number: 1184259
zbMATH Keywords: generic large cardinals
zbMATH Keywords: research survey
zbMATH Keywords: elementary embeddings
zbMATH Keywords: forcing extensions
reviewed by: Pierre Matet
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1184259