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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.

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Se basa en

Axiomatic set theory: hierarchy, cardinal arithmetic and L

Requisitos previos obligatorios, hasta el fondo: 24

Aprenderás a

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.

Definición

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.

Teorema

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.

Errores frecuentes

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.

Conduce a

Large cardinals

Descubrimientos opcionales

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

Lecturas complementarias

Kenneth Kunen, Set Theory: An Introduction to Independence Proofs (1980)
Paul J. Cohen, Set Theory and the Continuum Hypothesis (1966)
Thomas Jech, Set Theory (Third Millennium Edition) (2003)