Virtual Models in Vλ

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Table of contents

Introduction
1 Preliminaries
1.1 Approachability Ideal
1.2 Models and Approximation Property
1.3 Strong Properness
1.4 Forcing Axioms
1.5 Large Cardinals
2 Warm-up
2.1 Scaffolding Posets
3 The Theory of Virtual Models
3.1 General Virtual Models
3.2 Virtual Models in Vλ
4 Forcing with Virtual Models of Two Types
4.1 Pure Side Conditions without Decorations
4.2 Pure Side Conditions with Decorations
4.3 Adding CLUBs
4.4 Quotients
5 Iteration and Forcing Axiom
5.1 Virtual Model Based Iteration
5.2 Quotients by Magidor Models
6 Guessing Models
6.1 Guessing Models
6.2 Consistency of GM+(ω3, ω1)
6.3 Consistency of SGM+(ω3, ω1)
A A Schematic Diagram of the Principles
B MRP with Finite Conditions
C Open Problems
Bibliography