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Maths › 03. Logic and Foundations ›

9.1 Forcing

Forcing is a method for constructing new models of set theory from old ones by adjoining new sets in a controlled way, and its main purpose is to show that certain statements cannot be decided from a given list of axioms, such as ZFC, provided those axioms are consistent. The guiding idea is that we begin with a model $M$ of set theory and a partially ordered set $\mathbb{P}$ whose...
logic set theory forcing independence May 2, 2026
Maths › 03. Logic and Foundations ›

9. Advanced Set Theory

Advanced set theory studies methods and principles that go beyond the basic axioms of ZF and ZFC, especially when ordinary axioms do not decide important mathematical questions. The chapter begins with forcing, which is a method for constructing new models of set theory from old ones, and it provides the main technique for proving independence results such as the independence of the continuum hypothesis. Large cardinals are then introduced as...
logic set theory forcing large cardinals descriptive set theory May 2, 2026
Maths › 03. Logic and Foundations ›

8.5 Independence Phenomena

An independence result shows that a given statement cannot be proved or refuted from a given system of axioms, assuming that the system itself is consistent. In set theory, many natural and important statements turn out to be independent of the standard axioms. The study of independence phenomena reveals that the axioms of ZF and ZFC do not determine a single complete picture of the universe of sets. Instead, different...
logic set theory independence forcing constructibility May 2, 2026