forcing (set theory)
E446865
Forcing (set theory) is a powerful technique in mathematical logic, introduced by Paul Cohen, used to construct models of set theory and prove the independence of certain propositions from Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC).
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
method in set theory
ⓘ
model construction method ⓘ technique in mathematical logic ⓘ |
| appliesTo |
models of ZFC
ⓘ
transitive models of set theory ⓘ |
| basedOn |
Boolean-valued models
ⓘ
partial orders ⓘ |
| coreConcept |
dense subset
ⓘ
forcing condition ⓘ genericity over a ground model ⓘ names (forcing names) ⓘ |
| defines |
forcing extension
ⓘ
forcing notion ⓘ forcing relation ⓘ generic filter ⓘ |
| enables |
construction of models where the Continuum Hypothesis fails
ⓘ
construction of models where the Continuum Hypothesis holds ⓘ construction of models with different cardinal arithmetic ⓘ construction of models with or without certain combinatorial principles ⓘ construction of models with special subsets of the reals ⓘ |
| field |
mathematical logic
ⓘ
set theory ⓘ |
| goal |
control which new sets are added to a model
ⓘ
extend a ground model to a larger model ⓘ |
| hasVariant |
Martin’s Axiom related forcing
ⓘ
class forcing ⓘ forcing with side conditions ⓘ iterated forcing ⓘ proper forcing ⓘ semi-proper forcing ⓘ symmetric submodel forcing ⓘ |
| historicalPeriod | 1960s ⓘ |
| influenced |
modern set-theoretic research
ⓘ
philosophy of set-theoretic truth ⓘ |
| introducedBy | Paul Cohen NERFINISHED ⓘ |
| notableResult |
independence of Suslin’s Hypothesis from ZFC
ⓘ
independence of the Axiom of Choice from some weaker systems ⓘ independence of the Continuum Hypothesis from ZFC ⓘ |
| relatedTo |
Boolean-valued models of set theory
NERFINISHED
ⓘ
descriptive set theory ⓘ inner model theory ⓘ large cardinal axioms ⓘ |
| standardReference |
Set Theory by Kenneth Kunen
NERFINISHED
ⓘ
Set Theory by Thomas Jech NERFINISHED ⓘ |
| usedFor |
constructing models of set theory
ⓘ
proving independence results ⓘ proving the independence of Suslin’s Hypothesis ⓘ proving the independence of the Axiom of Choice variants ⓘ proving the independence of the Continuum Hypothesis ⓘ proving the independence of the existence of measurable cardinals from weaker theories ⓘ showing consistency of statements relative to ZFC ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.