An Introduction to Core Model Theory

Summary

Abstract. In this paper we give an informal introduction to core model theory at the level of Woodin cardinals.

Introduction

Zermelo-Praenkel set theory with choice, or ZFC, is the commonly accepted system of axioms for set theory, and hence for all of mathematics. Most of the axioms of ZFC express closure properties of the universe of sets. (The exceptions are Extensionality and Foundation, which in effect limit the objects under consideration.) Although all mathematical assertions can be expressed in the language of ZFC, and “most” of them can be decided using only the axioms of ZFC, there are nevertheless interesting mathematical assertions which cannot be decided using ZFC alone. The most famous of these is the Continuum Hypothesis.

Gödel's response to the incompleteness of ZFC with respect to assertions like the Continuum Hypothesis was that one should seek well–justified extensions of ZFC which decide these assertions (cf. [Gö47]). This is known as “Gödel's Program” and is still one of the most important tasks of higher set theory. Gödel suggested strong axioms of infinity, now more commonly known as large cardinal axioms, as candidates for basic principles to be added to the foundation provided by ZFC. In the years since [GÖ47], large cardinal axioms have been extensively investigated, and have proved very fruitful in deciding in natural ways propositions about the real numbers left undecided by ZFC. They do not decide the Continuum Hypothesis, however.