First-year graduateFoundationsPlanned

Forcing and independence

Relative consistency and independence: countable transitive models, posets of conditions, generic filters and the forcing relation, leading to Cohen's proof that the continuum hypothesis is independent of ZFC.

math.found.forcing

Show everything needed to reach this topic

The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.

Builds on

Axiomatic set theory: hierarchy, cardinal arithmetic and L

Required prerequisites, all the way down: 23

You will learn to

Explain what it means for a statement to be independent of ZFC and why independence proofs are relative-consistency proofs.
Construct a generic extension M[G] by Cohen forcing and verify that it satisfies ZFC + ¬CH, given a countable transitive model M of ZFC.
State Easton's theorem and give other examples of statements shown independent by forcing, such as Suslin's hypothesis.

Definition

A statement φ is independent of ZFC if neither φ nor ¬φ is provable from ZFC.
A forcing notion is a poset (P, ≤) of conditions; a filter G ⊆ P is M-generic if it meets every dense subset of P that belongs to M.
The forcing relation p ⊩ φ means that φ holds in every generic extension M[G] with p ∈ G.

Theorem

Cohen (1963): if ZFC is consistent, so is ZFC + ¬CH. Together with Gödel's work on L, CH is independent of ZFC.
Truth lemma: φ holds in M[G] if and only if some p ∈ G forces φ.
Easton (1970): on regular cardinals, the function κ ↦ 2^κ can consistently be any non-decreasing function F with cf(F(κ)) > κ.

Common misconceptions

Thinking independence makes a question meaningless; it means ZFC does not settle it.
Believing forcing requires a countable transitive model of ZFC to exist; formal treatments avoid this with Boolean-valued models or by working with finite fragments of ZFC.

Leads to

Large cardinals

Optional discoveries

Descriptive set theory Many independent statements concern sets of reals, where descriptive set theory gives a finer classification of which sets are well behaved.
Elementary topos theory Cohen's construction can be presented as passing to a topos of sheaves on a poset of conditions, as in Mac Lane and Moerdijk's treatment of the independence of CH.
← Countability and Cantor's diagonal argument
← Ordinals and cardinals
← Model theory