What is a generic filter intuitively?

It is an idealized object chosen to meet every requirement that is definable in the ground model, so that it is sufficiently generic to avoid being captured by any single definition there. Adjoining it produces a controlled extension of the universe of sets.

Does forcing change the truth of the axioms of set theory?

No. A generic extension of a model of ZFC is again a model of ZFC; forcing changes only the truth value of statements left undetermined by the axioms, such as the continuum hypothesis, which is exactly what makes it a tool for independence proofs.

Forcing and Independence

Forcing is a technique for extending a model of set theory by adjoining a carefully chosen generic object, and it is the principal method for proving that statements are independent of the standard axioms.

Definition

Forcing is a method that, starting from a model of set theory and a partial order within it, builds a larger model containing a generic filter; by controlling the partial order one arranges that prescribed statements hold or fail in the extension, thereby proving their consistency or independence.

Scope

This topic covers the method of forcing, partial orders and generic filters, the forcing relation and the construction of generic extensions, the preservation of cardinals via chain conditions, and the canonical independence results for the continuum hypothesis and the axiom of choice, together with Goedel's complementary constructible universe.

Core questions

How does adjoining a generic filter produce a new model of set theory?
How is truth in the generic extension controlled by the forcing relation within the ground model?
Which combinatorial properties of the forcing poset preserve cardinals and cofinalities?
How do forcing and the constructible universe together establish the independence of the continuum hypothesis?

Key theories

Generic extensions and the forcing theorem: Given a generic filter over a partial order, every statement true in the resulting extension is forced by some condition, and this forcing relation is definable in the ground model, allowing the extension to be analyzed from within.
Constructible universe and consistency of CH: Goedel's inner model of constructible sets satisfies the axiom of choice and the generalized continuum hypothesis, showing these are consistent with the other axioms.
Independence of the continuum hypothesis: Cohen used forcing to add many reals to a model so that the continuum hypothesis fails, which together with Goedel's result shows the hypothesis is independent of ZFC.

Clinical relevance

Forcing is the central tool of contemporary set theory: it is used to prove the independence of a wide range of statements in analysis, topology, and algebra, and to calibrate the strength of combinatorial principles, revealing which mathematical questions the standard axioms cannot settle.

History

Goedel introduced the constructible universe in 1938 to prove the consistency of the continuum hypothesis and the axiom of choice. In 1963 Cohen invented forcing to prove their independence, work for which he received the Fields Medal; Scott, Solovay, and others reformulated forcing via Boolean-valued models and developed it into the standard apparatus of the field.

Key figures

Paul Cohen
Kurt Goedel
Dana Scott
Robert Solovay

Seminal works

kunen2011
cohen1963
godel1940

Frequently asked questions

What is a generic filter intuitively?: It is an idealized object chosen to meet every requirement that is definable in the ground model, so that it is sufficiently generic to avoid being captured by any single definition there. Adjoining it produces a controlled extension of the universe of sets.
Does forcing change the truth of the axioms of set theory?: No. A generic extension of a model of ZFC is again a model of ZFC; forcing changes only the truth value of statements left undetermined by the axioms, such as the continuum hypothesis, which is exactly what makes it a tool for independence proofs.