# What is the Continuum Hypothesis?

• 1.2 Current Status of CH Alternate Overview • Assumptions, Style, and Terminology • 2.1 Assumptions • 2.1.1 Audience Assumptions • 2.1.2 Mathematical Assumptions • 2.2 Style • 2.3 Terminology • 2.3.1 The Word “continuum” • 2.3.2 Ordered Sets • 2.3.3 More Terms and Notation • Mathematics of the Continuum and CH • 3.1 Sizes of Sets: Cardinal Numbers • 3.1.1 aleph0 < c = 2aleph0 • 3.1.2 CH and GCH • 3.1.3 Sample Cardinalities • 3.2 Ordering Sets: Ordinal Numbers • 3.3 Analysis of the Continuum • 3.3.1 Decomposing the Reals • 3.3.2 Characterizing the Reals • 3.3.3 Characterizing Continuity • 3.4 What ZFC Does and Does Not Tell Us About c • Metamathematics and CH • 4.1 Consistency, Completeness, and Compactness of ... • 4.1.1 a Logical System • 4.1.2 an Axiomatic Theory • 4.2 Models of ... • 4.2.1 Real Numbers • 4.2.2 Set Theory • 4.2.2.1 Inner Models • 4.2.2.2 Forcing and Outer Models • 4.3 Adding Axioms to Zermelo Fraenkel Set Theory • 4.3.1 Axioms that Imply CH or GCH • 4.3.1.1 Explici