The higher infinite : large cardinals in set theory from their beginnings

  title={The higher infinite : large cardinals in set theory from their beginnings},
  author={Akihiro Kanamori},
The theory of large cardinals is currently a broad mainstream of modern set theory, the main area of investigation for the analysis of the relative consistency of mathematical propositions and possible new axioms for mathematics. The first of a projected multi-volume series, this book provides a comprehensive account of the theory of large cardinals from its beginnings and some of the direct outgrowths leading to the frontiers of contemporary research. A "genetic" approach is taken, presenting… 
Large Cardinals with Forcing
  • A. Kanamori
  • Biology
    Sets and Extensions in the Twentieth Century
  • 2012
The Landscape of Large Cardinals
. The purpose of this paper is to provide an introductory overview of the large cardinal hierarchy in set theory. By a large cardinal, we mean any cardinal κ whose existence is strong enough of an
Large Cardinals and Determinacy
The developments of set theory in 1960’s led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC.
Independence and Large Cardinals
The manner in which large cardinal axioms provide a canonical means for climbing the hierarchy of interpretability and serve as an intermediary in the comparison of systems from conceptually distinct domains is discussed.
Successor Large Cardinals in Symmetric Extensions∗
We give an exposition in modern language (and using partial orders) of Jech’s method for obtaining models where successor cardinals have large cardinal properties. In such models, the axiom of choice
Executing Gödel’s programme in set theory
The study of set theory (a mathematical theory of infinite collections) has garnered a great deal of philosophical interest since its development. There are several reasons for this, not least
How Gödel Transformed Set Theory
K urt Gödel (1906–1978), with his work on the constructible universe L, established the relative consistency of the Axiom of Choice and the Continuum Hypothesis. More broadly, he secured the
The cofinality of the least Berkeley cardinal and the extent of dependent choice
This paper is concerned with the possible values of the cofinality of the least Berkeley cardinal. Berkeley cardinals are very large cardinal axioms incompatible with the Axiom of Choice, and the
Beginning Inner Model Theory
This chapter provides an introduction to the basic theory of inner models of set theory, without fine structure. Section 1 begins with the basic theory of Godel’s class L of constructible sets, with
Large sets in constructive set theory
This thesis presents an investigation into large sets and large set axioms in the context of the constructive set theory CZF. We determine the structure of large sets by classifying their von